Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682648 |
| 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 | #60770 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1747 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)(3,6)(4,5)(7,8)(9,10,16,21)(11,17,23,39)(12,26,30,45)(13,19,33,50)(14,34,15,20)(18,36,46,29)(22,43,51,64)(24,40,54,63)(25,42,35,41)(27,56,53,38)(28,57,61,68)(31,47,55,65)(32,49,37,48)(44,60,67,59)(52,66,69,72)(58,71,70,62);(2,5)(3,7)(9,11,16,23)(10,17,21,39)(12,27,30,53)(13,31,33,55)(14,25,15,35)(18,43,46,64)(19,47,50,65)(20,41,34,42)(22,36,51,29)(24,37,54,32)(26,56,45,38)(28,52,61,69)(40,48,63,49)(44,62,67,71)(57,66,68,72)(58,59,70,60);(9,12,15,36,16,30,14,29)(10,18,34,45,21,46,20,26)(11,22,35,27,23,51,25,53)(13,28,37,60,33,61,32,59)(17,38,42,64,39,56,41,43)(19,44,49,68,50,67,48,57)(24,52,31,58,54,69,55,70)(40,62,65,72,63,71,47,66);(2,5)(6,8)(11,23)(17,39)(22,51)(24,54)(25,35)(27,53)(31,55)(38,56)(40,63)(41,42)(43,64)(47,65)(52,69)(58,70)(62,71)(66,72);(1,3,4,7)(2,6,5,8)(9,13)(10,19)(11,24)(12,28)(14,32)(15,37)(16,33)(17,40)(18,44)(20,48)(21,50)(22,52)(23,54)(25,55)(26,57)(27,58)(29,59)(30,61)(31,35)(34,49)(36,60)(38,62)(39,63)(41,47)(42,65)(43,66)(45,68)(46,67)(51,69)(53,70)(56,71)(64,72)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=gi,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,h^b=hi,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, 6592703);