Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7624184 |
| 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 | |
1493 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)(13,20)(14,32)(15,36)(16,19)(18,28)(21,34)(22,48)(23,42)(24,39)(25,55)(27,58)(29,57)(30,41)(31,46)(33,44)(35,61)(37,47)(38,51)(40,54)(43,59)(45,62)(49,69)(50,66)(52,72)(53,65)(56,68)(60,67)(63,71)(64,70);(1,3)(2,5)(4,8)(6,7)(9,11,16,23)(10,17,19,42)(12,22,29,49)(13,30,15,24)(14,25,35,53)(18,38,43,64)(20,41,36,39)(21,40,46,67)(26,48,57,69)(27,52,37,50)(28,51,59,70)(31,60,34,54)(32,55,61,65)(33,56,62,71)(44,68,45,63)(47,66,58,72);(9,12)(10,18)(11,22)(13,27)(14,33)(15,37)(16,29)(17,38)(19,43)(20,44)(21,47)(23,49)(24,50)(25,56)(26,32)(28,34)(30,52)(31,59)(35,62)(36,45)(39,63)(40,66)(41,68)(42,64)(46,58)(48,55)(51,54)(53,71)(57,61)(60,70)(65,69)(67,72);(3,8)(5,7)(10,19)(11,23)(18,43)(20,36)(21,46)(22,49)(24,30)(25,53)(26,57)(32,61)(44,45)(47,58)(50,52)(51,70)(54,60)(56,71);(2,6)(5,7)(9,13,16,15)(10,20,19,36)(11,24,23,30)(12,27,29,37)(14,31,35,34)(17,39,42,41)(18,44,43,45)(21,32,46,61)(22,50,49,52)(25,54,53,60)(26,58,57,47)(28,33,59,62)(38,63,64,68)(40,65,67,55)(48,66,69,72)(51,71,70,56)Polycyclic presentation:
2^9|b^2=i,e^2=i,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,g^a=gh,h^a=hi,d^b=dfi,e^b=ei,g^c=gh,h^c=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531392);