Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682658 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #61216 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1783 conjugacy classes of subgroups.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 72 points:
(1,2,5,7)(3,8,4,6)(9,10,16,21)(11,17,25,42)(12,26,28,46)(13,19,33,50)(14,34,15,20)(18,36,44,30)(22,43,51,64)(23,39,55,63)(24,41,35,40)(27,56,53,38)(29,57,59,68)(31,47,54,65)(32,49,37,48)(45,61,67,60)(52,66,69,72)(58,71,70,62);(1,2)(3,6)(4,8)(5,7)(9,11)(10,17)(12,27)(13,31)(14,24)(15,35)(16,25)(18,43)(19,47)(20,40)(21,42)(22,30)(23,32)(26,56)(28,53)(29,52)(33,54)(34,41)(36,51)(37,55)(38,46)(39,49)(44,64)(45,62)(48,63)(50,65)(57,66)(58,61)(59,69)(60,70)(67,71)(68,72);(9,12,14,30,16,28,15,36)(10,18,20,26,21,44,34,46)(11,22,24,53,25,51,35,27)(13,29,32,60,33,59,37,61)(17,38,40,43,42,56,41,64)(19,45,48,57,50,67,49,68)(23,52,54,70,55,69,31,58)(39,62,47,66,63,71,65,72);(1,3,5,4)(2,6,7,8)(12,28)(18,44)(22,51)(26,46)(27,53)(29,59)(30,36)(38,56)(43,64)(45,67)(52,69)(57,68)(58,70)(60,61)(62,71)(66,72);(1,4,5,3)(2,6,7,8)(9,13)(10,19)(11,23)(12,29)(14,32)(15,37)(16,33)(17,39)(18,45)(20,48)(21,50)(22,52)(24,54)(25,55)(26,57)(27,58)(28,59)(30,60)(31,35)(34,49)(36,61)(38,62)(40,47)(41,65)(42,63)(43,66)(44,67)(46,68)(51,69)(53,70)(56,71)(64,72)Polycyclic presentation:
2^9|a^2=fi,c^2=g,d^2=f,e^2=f,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, 6594188);