Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7624888 |
| 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 | |
1368 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)(9,10,16,21)(11,17,24,39)(12,27,30,57)(13,19,15,36)(14,34,35,61)(18,41,43,31)(20,33,45,28)(22,46,50,69)(23,38,26,55)(25,54,53,72)(29,56,32,58)(37,62,65,49)(40,52,66,47)(42,60,44,59)(48,67,51,68)(63,71,64,70);(1,2)(3,6)(4,5)(7,8)(9,11)(10,17)(12,22)(13,23)(14,25)(15,26)(16,24)(18,37)(19,38)(20,40)(21,39)(27,46)(28,47)(29,48)(30,50)(31,49)(32,51)(33,52)(34,54)(35,53)(36,55)(41,62)(42,63)(43,65)(44,64)(45,66)(56,67)(57,69)(58,68)(59,70)(60,71)(61,72);(9,12,28,59,15,29,35,41,16,30,33,60,13,32,14,31)(10,18,34,58,19,44,45,57,21,43,61,56,36,42,20,27)(11,22,47,70,26,48,53,62,24,50,52,71,23,51,25,49)(17,37,54,68,38,64,66,69,39,65,72,67,55,63,40,46);(2,5)(6,8)(9,13,16,15)(10,19,21,36)(11,23,24,26)(12,29,30,32)(14,33,35,28)(17,38,39,55)(18,42,43,44)(20,34,45,61)(22,48,50,51)(25,52,53,47)(27,56,57,58)(31,59,41,60)(37,63,65,64)(40,54,66,72)(46,67,69,68)(49,70,62,71);(1,3,4,7)(2,6,5,8)(11,24)(12,30)(17,39)(18,43)(23,26)(25,53)(27,57)(29,32)(31,41)(38,55)(40,66)(42,44)(47,52)(54,72)(56,58)(59,60)Polycyclic presentation:
2^9|a^2=i,c^2=gh,d^2=i,e^2=f,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gh,h^a=hi,d^b=df,e^b=ei,d^c=di,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532223);