Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7624170 |
| 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 ◁ #525 ◁ #2929 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1506 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 72 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,17)(12,26)(13,19)(14,32)(15,36)(16,21)(18,28)(20,34)(22,48)(23,39)(24,40)(25,55)(27,58)(29,59)(30,42)(31,47)(33,43)(35,57)(37,46)(38,51)(41,53)(44,62)(45,60)(49,67)(50,69)(52,72)(54,65)(56,68)(61,66)(63,70)(64,71);(1,3)(2,5)(4,8)(6,7)(9,11)(10,17)(12,22)(13,30)(14,25)(15,23)(16,24)(18,38)(19,42)(20,41)(21,40)(26,48)(27,52)(28,51)(29,50)(31,61)(32,55)(33,56)(34,53)(35,54)(36,39)(37,49)(43,68)(44,63)(45,64)(46,67)(47,66)(57,65)(58,72)(59,69)(60,71)(62,70);(9,12,16,29)(10,18,21,45)(11,22,24,50)(13,27,15,37)(14,33,35,62)(17,38,40,64)(19,43,36,44)(20,46,47,58)(23,49,30,52)(25,56,54,70)(26,57,59,32)(28,31,60,34)(39,63,42,68)(41,67,66,72)(48,65,69,55)(51,61,71,53);(3,8)(5,7)(9,13,16,15)(10,19,21,36)(11,23,24,30)(12,27,29,37)(14,31,35,34)(17,39,40,42)(18,43,45,44)(20,32,47,57)(22,49,50,52)(25,53,54,61)(26,58,59,46)(28,33,60,62)(38,63,64,68)(41,65,66,55)(48,67,69,72)(51,70,71,56);(2,6)(5,7)(11,24)(17,40)(22,50)(23,30)(25,54)(38,64)(39,42)(41,66)(48,69)(49,52)(51,71)(53,61)(55,65)(56,70)(63,68)(67,72)Polycyclic presentation:
2^9|c^2=i,d^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gh,h^a=hi,d^b=dfi,e^b=ei,g^c=gh,h^c=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531374);