Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150144 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2140 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C22×C16 | C23 |
| socle | C2 | #12813 |
| prime radical | C2×#508 | C22 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ #405 ◁ #2108 ◁ G | |
| LECS | G ▷ C22×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
258 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 258 conjugacy classes of subgroups — too many to draw. Only its 88 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 128 points:
(1,2,6,14,9,12,8,40)(3,16,19,46,21,10,35,44)(4,22,25,75,23,77,41,76)(5,27,31,83,34,82,33,95)(7,37,36,53,39,98,30,15)(11,47,48,78,51,92,50,24)(13,54,57,94,52,93,84,32)(17,61,64,119,62,101,66,123)(18,67,70,124,28,85,88,125)(20,72,69,60,74,97,87,45)(26,81,38,56,79,116,99,113)(29,58,91,49,90,111,96,112)(42,100,102,63,59,110,109,80)(43,104,106,71,55,114,107,89)(65,122,121,105,120,128,73,115)(68,118,126,103,86,108,127,117);(1,3,8,19,9,21,6,35)(2,10,40,44,12,16,14,46)(4,17,41,64,23,62,25,66)(5,28,33,88,34,18,31,70)(7,20,30,69,39,74,36,87)(11,42,50,102,51,59,48,109)(13,55,84,107,52,43,57,106)(15,45,98,72,53,60,37,97)(22,61,76,119,77,101,75,123)(24,80,92,100,78,63,47,110)(26,65,99,121,79,120,38,73)(27,67,95,124,82,85,83,125)(29,86,96,127,90,68,91,126)(32,89,93,104,94,71,54,114)(49,103,58,108,112,117,111,118)(56,115,81,122,113,105,116,128);(1,4)(2,11)(3,17)(5,29)(6,25)(7,38)(8,41)(9,23)(10,42)(12,51)(13,56)(14,48)(15,58)(16,59)(18,68)(19,64)(20,73)(21,62)(22,27)(24,32)(26,30)(28,86)(31,91)(33,96)(34,90)(35,66)(36,79)(37,49)(39,99)(40,50)(43,105)(44,102)(45,108)(46,109)(47,54)(52,113)(53,111)(55,115)(57,116)(60,118)(61,67)(63,71)(65,69)(70,126)(72,117)(74,121)(75,83)(76,95)(77,82)(78,94)(80,89)(81,84)(85,101)(87,120)(88,127)(92,93)(97,103)(98,112)(100,104)(106,128)(107,122)(110,114)(119,124)(123,125)Polycyclic presentation:
2^9|a^2=f,b^2=dh,e^2=gh,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gh,h^a=hi,e^b=ei,f^b=fi,d^c=di,e^c=eg,g^c=gh,h^c=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72320367, #72320842, #72320903 |
| maximal normal subgroup of | #72320367, #72320842, #72320903 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59484);