Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7622979 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1354 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 72 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,23)(12,27)(13,18)(14,20)(15,21)(16,38)(17,39)(19,32)(22,28)(24,49)(25,43)(26,53)(29,54)(30,55)(31,56)(33,64)(34,41)(35,65)(36,46)(37,66)(40,61)(42,59)(44,62)(45,63)(47,52)(48,50)(51,69)(57,58)(60,70)(67,72)(68,71);(2,5)(6,8)(9,11,15,13)(10,18,21,23)(12,24,31,29)(14,34,17,25)(16,26,37,35)(19,40,45,47)(20,43,39,41)(22,42,46,48)(27,54,56,49)(28,50,36,59)(30,60,33,51)(32,52,63,61)(38,65,66,53)(44,68,57,67)(55,69,64,70)(58,71,62,72);(9,12,28,58,17,33,37,63,15,31,36,62,14,30,16,32)(10,19,38,55,20,44,46,56,21,45,66,64,39,57,22,27)(11,24,50,71,25,51,35,61,13,29,59,72,34,60,26,52)(18,40,65,69,43,68,48,49,23,47,53,70,41,67,42,54);(1,3,4,7)(2,6,5,8)(9,13,15,11)(10,18,21,23)(12,29,31,24)(14,25,17,34)(16,35,37,26)(19,40,45,47)(20,43,39,41)(22,42,46,48)(27,54,56,49)(28,59,36,50)(30,51,33,60)(32,61,63,52)(38,65,66,53)(44,68,57,67)(55,69,64,70)(58,72,62,71);(9,14,15,17)(10,20,21,39)(11,25,13,34)(12,30,31,33)(16,36,37,28)(18,41,23,43)(19,44,45,57)(22,38,46,66)(24,51,29,60)(26,50,35,59)(27,55,56,64)(32,62,63,58)(40,67,47,68)(42,53,48,65)(49,69,54,70)(52,71,61,72)Polycyclic presentation:
2^9|b^2=i,c^2=gh,d^2=f,e^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,g^a=gh,h^a=hi,e^b=ei,e^d=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530468);