Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682658 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1876 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 72 points:
(1,2,4,5)(3,7,8,6)(9,10,14,19)(11,22,25,17)(12,27,29,46)(13,20,33,47)(15,35,16,21)(18,37,45,31)(23,53,54,43)(24,52,57,40)(26,41,36,42)(28,39,56,51)(30,59,61,71)(32,65,58,48)(34,50,38,49)(44,64,70,63)(55,72,62,68)(60,67,66,69);(1,2)(3,6)(4,5)(7,8)(9,11)(10,17)(12,28)(13,32)(14,25)(15,26)(16,36)(18,43)(19,22)(20,48)(21,41)(23,31)(24,38)(27,51)(29,56)(30,62)(33,58)(34,57)(35,42)(37,54)(39,46)(40,49)(44,69)(45,53)(47,65)(50,52)(55,61)(59,72)(60,63)(64,66)(67,70)(68,71);(2,5)(6,7)(9,12,16,37,14,29,15,31)(10,18,35,46,19,45,21,27)(11,23,36,28,25,54,26,56)(13,30,38,64,33,61,34,63)(17,39,42,53,22,51,41,43)(20,44,50,71,47,70,49,59)(24,55,58,60,57,62,32,66)(40,67,48,72,52,69,65,68);(1,3,4,8)(2,6,5,7)(10,19)(12,29)(17,22)(20,47)(21,35)(23,54)(28,56)(30,61)(31,37)(40,52)(41,42)(48,65)(49,50)(55,62)(60,66)(63,64);(9,13)(10,20)(11,24)(12,30)(14,33)(15,34)(16,38)(17,40)(18,44)(19,47)(21,49)(22,52)(23,55)(25,57)(26,32)(27,59)(28,60)(29,61)(31,63)(35,50)(36,58)(37,64)(39,67)(41,65)(42,48)(43,68)(45,70)(46,71)(51,69)(53,72)(54,62)(56,66)Polycyclic presentation:
2^9|a^2=f,c^2=fg,d^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6592182);