Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 7 |
| isoclinism | isoclinic class of #94835 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7537 |
| commutator | C64 | C2×C4 |
| Frattini, cosocle | C2×C64 | C22 |
| prime radical | C2×D64 | C2 |
| LCS | G ▷ C64 ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ C2×C32 ◁ C2×C64 ◁ G | |
| LECS | G ▷ C2×C64 ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
55 conjugacy classes of subgroups.
| minimal degree | μ(G) = 128 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 128 points:
(1,2,4,12,9,16,8,41)(3,17,20,53,25,10,24,42)(5,29,26,86,34,48,33,97)(6,35,27,87,38,99,37,100)(7,39,28,62,40,102,36,15)(11,46,49,18,52,66,103,21)(13,56,54,19,47,70,104,30)(14,58,55,93,60,88,98,31)(22,78,73,120,82,59,81,123)(23,83,74,45,85,61,80,65)(32,94,71,114,96,128,91,90)(43,107,105,76,63,111,118,68)(44,109,106,75,64,110,119,67)(50,115,108,79,116,127,124,126)(51,77,101,84,117,69,122,125)(57,92,89,121,112,72,113,95);(2,10)(3,18)(4,8)(5,30)(6,31)(7,36)(11,47)(12,53)(13,52)(14,59)(15,61)(16,17)(19,34)(20,46)(21,25)(22,79)(23,84)(24,66)(26,70)(27,88)(28,40)(29,43)(32,95)(33,56)(35,44)(37,58)(38,93)(39,45)(41,42)(48,63)(49,54)(50,114)(51,113)(55,120)(57,122)(60,78)(62,83)(64,99)(65,102)(67,76)(68,75)(69,80)(71,72)(73,115)(74,77)(81,127)(82,126)(85,125)(86,118)(87,119)(89,117)(90,116)(91,92)(94,108)(96,121)(97,105)(98,123)(100,106)(101,112)(103,104)(107,109)(110,111)(124,128)Polycyclic presentation:
2^9|a^2=d,c^2=efi,d^2=i,e^2=fg,f^2=gh,g^2=hi,h^2=i,b^a=bc,c^a=ce,e^a=ef,f^a=fg,g^a=gh,h^a=hi,c^b=cei,d^b=di,e^b=ef,f^b=fg,g^b=gh,h^b=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72230770 |
| maximal normal subgroup of | #72230770 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2032);