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 ◁ #529 ◁ #2966 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1288 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 72 points:
(2,5)(3,7)(9,10)(11,23)(12,29)(13,22)(14,20)(15,35)(16,19)(17,26)(18,32)(21,37)(24,51)(25,40)(27,57)(28,42)(30,63)(31,62)(33,46)(34,45)(36,48)(38,47)(39,67)(41,55)(43,65)(44,64)(49,54)(50,59)(52,69)(53,61)(56,66)(58,68)(60,71)(70,72);(1,2)(3,6)(4,5)(7,8)(9,11)(10,17)(12,24)(13,25)(14,26)(15,27)(16,28)(18,39)(19,40)(20,23)(21,41)(22,42)(29,61)(30,52)(31,53)(32,54)(33,55)(34,56)(35,66)(36,58)(37,59)(38,60)(43,68)(44,49)(45,57)(46,50)(47,69)(48,70)(51,62)(63,71)(64,67)(65,72);(9,12,14,31)(10,18,20,44)(11,24,26,53)(13,30,16,38)(15,36,34,65)(17,39,23,49)(19,43,22,48)(21,47,46,63)(25,52,28,60)(27,58,56,72)(29,45,62,35)(32,33,64,37)(40,68,42,70)(41,69,50,71)(51,66,61,57)(54,55,67,59);(2,5)(6,8)(9,13,14,16)(10,19,20,22)(11,25,26,28)(12,30,31,38)(15,33,34,37)(17,40,23,42)(18,43,44,48)(21,45,46,35)(24,52,53,60)(27,55,56,59)(29,47,62,63)(32,36,64,65)(39,68,49,70)(41,57,50,66)(51,71,61,69)(54,58,67,72);(1,3,4,7)(2,6,5,8)(10,20)(11,26)(18,44)(19,22)(21,46)(24,53)(25,28)(27,56)(29,62)(35,45)(43,48)(47,63)(52,60)(54,67)(55,59)(58,72)Polycyclic presentation:
2^9|c^2=i,d^2=i,e^2=fi,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, 7531846);