Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7624842 |
| 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 | |
1271 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)(9,10,15,20)(11,16,22,36)(12,25,28,54)(13,31,33,63)(14,34,32,19)(17,39,41,29)(18,43,44,27)(21,45,26,53)(23,51,49,67)(24,52,50,38)(30,62,61,56)(35,58,40,47)(37,57,65,46)(42,60,55,66)(48,71,59,69)(64,70,68,72);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,26)(13,23)(14,24)(15,22)(17,40)(18,37)(19,38)(20,36)(21,28)(25,53)(27,46)(29,58)(30,59)(31,51)(32,50)(33,49)(34,52)(35,41)(39,47)(42,68)(43,57)(44,65)(45,54)(48,61)(55,64)(56,71)(60,72)(62,69)(63,67)(66,70);(9,12,27,60,14,30,33,39,15,28,43,66,32,61,13,29)(10,17,31,56,19,42,44,54,20,41,63,62,34,55,18,25)(11,21,46,70,24,48,49,58,22,26,57,72,50,59,23,47)(16,35,51,69,38,64,65,53,36,40,67,71,52,68,37,45);(3,8)(5,7)(12,28)(17,41)(21,26)(25,54)(29,39)(30,61)(35,40)(42,55)(45,53)(47,58)(48,59)(56,62)(60,66)(64,68)(69,71)(70,72);(2,6)(5,7)(11,22)(16,36)(21,26)(23,49)(24,50)(35,40)(37,65)(38,52)(45,53)(46,57)(47,58)(48,59)(51,67)(64,68)(69,71)(70,72)Polycyclic presentation:
2^9|a^2=i,c^2=gh,g^2=hi,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gh,h^a=hi,c^b=ci,d^b=df,e^b=ei,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532045);