Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6814827 |
| 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 | |
1998 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)(11,22)(12,28)(13,19)(14,18)(15,21)(16,25)(17,32)(20,35)(23,53)(24,52)(26,40)(27,50)(29,38)(30,44)(31,61)(33,65)(34,66)(36,45)(37,49)(39,59)(41,54)(42,64)(43,63)(46,60)(47,67)(48,57)(51,56)(55,71)(58,70)(62,68)(69,72);(1,2)(3,6)(4,8)(5,7)(9,11)(10,16)(12,29)(13,33)(14,25)(15,26)(17,41)(18,22)(19,46)(20,40)(21,50)(23,32)(24,57)(27,35)(28,51)(30,56)(31,55)(34,58)(36,60)(37,59)(38,44)(39,48)(42,68)(43,53)(45,65)(47,69)(49,52)(54,63)(61,70)(62,67)(64,72)(66,71);(9,12,14,30)(10,17)(11,23)(13,34,36,31)(15,32,35,63)(16,38,22,51)(18,43)(19,47)(20,28)(21,44)(24,58)(25,54)(26,56)(27,29)(33,62)(37,67,57,64)(39,69,52,68)(40,41,50,53)(42,45)(46,70,65,71)(48,66)(49,61)(55,59)(60,72);(1,3,5,4)(2,6,7,8)(10,18)(12,30)(16,22)(19,45)(20,21)(23,54)(29,56)(31,34)(32,63)(39,52)(40,50)(46,65)(48,49)(55,58)(62,72)(64,67);(1,4,5,3)(2,6,7,8)(9,13,35,57,14,36,15,37)(10,19,20,48,18,45,21,49)(11,24,26,33,25,59,27,60)(12,31,63,67,30,34,32,64)(16,39,50,46,22,52,40,65)(17,42,28,61,43,47,44,66)(23,55,56,72,54,58,29,62)(38,68,53,71,51,69,41,70)Polycyclic presentation:
2^9|a^2=fi,c^2=gh,d^2=fi,e^2=fh,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,e^c=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6724161);