Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #150797 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #11858 |
| commutator | C2×C16 | C22×C4 |
| Frattini, cosocle | C2×M32 | C23 |
| prime radical | C23 | #509 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ #2108 ◁ G | |
| LECS | G ▷ C2×M32 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
188 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 188 conjugacy classes of subgroups — too many to draw. Only its 58 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)(3,16,19,61)(4,23)(5,30,33,59)(6,37)(7,40)(8,43)(9,15)(10,18)(11,27)(12,34,55,35)(13,32)(14,39)(17,63,66,118)(20,73,21,74)(22,75,44,62)(24,80,82,124)(25,83,85,126)(26,53)(28,91)(29,79)(31,57)(36,99,97,92)(38,102,41,54)(42,50)(45,93)(46,100)(47,94)(48,72)(49,108)(51,112,114,88)(52,84)(56,98,104,96)(58,71)(60,67,113,68)(64,103)(65,105)(69,123,90,109)(70,107)(76,81)(77,86,116,87)(78,89)(95,101)(106,120)(110,122)(111,121)(115,127)(117,128,125,119);(1,3,9,19)(2,10,15,48)(4,24,29,82)(5,31,33,70)(6,38,7,41)(8,44,39,22)(11,49,53,111)(12,54,55,102)(13,57,50,107)(14,58,43,47)(16,59,61,30)(17,64,66,84)(18,34,72,35)(20,32,21,42)(23,76,79,115)(25,65,85,122)(26,69,27,90)(28,67,89,68)(36,100,97,95)(37,101,40,46)(45,91,106,78)(51,113,114,60)(52,105,103,110)(56,74,104,73)(62,92,75,99)(63,117,118,125)(71,98,94,96)(77,109,116,123)(80,126,124,83)(81,86,127,87)(88,108,112,121)(93,119,120,128);(1,4,7,27,8,28,42,64,9,29,6,26,39,89,32,84)(2,11,50,79,14,52,40,91,15,53,13,23,43,103,37,78)(3,17,21,68,22,69,38,82,19,66,20,67,44,90,41,24)(5,25,35,87,36,88,98,119,33,85,34,86,97,112,96,128)(10,45,101,110,47,76,57,111,48,106,46,105,58,115,107,49)(12,51,59,116,56,83,99,117,55,114,30,77,104,126,92,125)(16,60,102,118,62,80,74,123,61,113,54,63,75,124,73,109)(18,65,31,93,71,121,95,127,72,122,70,120,94,108,100,81)Polycyclic presentation:
2^9|a^2=e,b^2=i,c^2=fi,d^2=ghi,f^2=h,g^2=h,h^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,g^a=ghi,h^a=hi,c^b=cfh,d^b=dg,e^b=efghi,f^b=fhi,g^b=ghi,h^b=hi,e^d=ei44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57995);