Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682648 |
| 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 |
| prime radical | #61083 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1878 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)(9,10,15,21)(11,16,26,41)(12,27,29,49)(13,18,34,46)(14,20,37,19)(17,32,43,31)(22,51,52,42)(23,39,54,64)(24,48,25,40)(28,38,55,62)(30,57,59,68)(33,45,56,65)(35,50,36,47)(44,61,67,60)(53,69,70,66)(58,63,71,72);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,28)(13,33)(14,24)(15,26)(17,42)(18,45)(19,40)(20,48)(21,41)(22,32)(23,36)(25,37)(27,38)(29,55)(30,53)(31,52)(34,56)(35,54)(39,47)(43,51)(44,63)(46,65)(49,62)(50,64)(57,69)(58,60)(59,70)(61,71)(66,68)(67,72);(2,5)(6,7)(9,12,14,31,15,29,37,32)(10,17,19,27,21,43,20,49)(11,22,24,28,26,52,25,55)(13,30,35,60,34,59,36,61)(16,38,40,51,41,62,48,42)(18,44,47,57,46,67,50,68)(23,53,33,58,54,70,56,71)(39,63,65,69,64,72,45,66);(1,3,4,8)(2,6,5,7)(12,29)(13,34)(17,43)(18,46)(22,52)(23,54)(27,49)(28,55)(31,32)(33,56)(35,36)(38,62)(39,64)(42,51)(45,65)(47,50);(9,13,15,34)(10,18,21,46)(11,23,26,54)(12,30,29,59)(14,35,37,36)(16,39,41,64)(17,44,43,67)(19,47,20,50)(22,53,52,70)(24,33,25,56)(27,57,49,68)(28,58,55,71)(31,60,32,61)(38,63,62,72)(40,65,48,45)(42,66,51,69)Polycyclic presentation:
2^9|a^2=fi,c^2=fg,d^2=f,e^2=i,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,e^d=ei,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6592189);