Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6814815 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2020 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)(11,17,25,42)(12,26,28,53)(13,19,33,47)(14,36,15,20)(18,31,44,30)(22,49,50,43)(23,39,52,65)(24,41,37,40)(27,38,55,63)(29,54,57,71)(32,46,61,66)(34,62,35,48)(45,60,68,59)(51,70,58,67)(56,64,72,69);(2,5)(3,7)(9,11)(10,17)(12,27)(13,32)(14,24)(15,37)(16,25)(18,43)(19,46)(20,40)(21,42)(22,31)(23,35)(26,38)(28,55)(29,58)(30,50)(33,61)(34,52)(36,41)(39,48)(44,49)(45,69)(47,66)(51,57)(53,63)(54,67)(56,60)(59,72)(62,65)(64,68)(70,71);(9,12,16,28)(10,18)(11,22)(13,29,33,57)(14,30,15,31)(17,38,42,63)(19,45)(20,26)(21,44)(23,51)(24,27)(25,50)(32,56)(34,59,35,60)(36,53)(37,55)(39,64,65,69)(40,49,41,43)(46,67,66,70)(47,68)(48,54)(52,58)(61,72)(62,71);(2,5)(6,8)(12,28)(13,33)(18,44)(19,47)(22,50)(23,52)(26,53)(27,55)(30,31)(32,61)(34,35)(38,63)(39,65)(43,49)(46,66)(48,62);(1,3,4,7)(2,6,5,8)(9,13,15,35,16,33,14,34)(10,19,20,48,21,47,36,62)(11,23,24,32,25,52,37,61)(12,29,31,60,28,57,30,59)(17,39,41,46,42,65,40,66)(18,45,26,54,44,68,53,71)(22,51,27,56,50,58,55,72)(38,64,43,67,63,69,49,70)Polycyclic presentation:
2^9|a^2=i,c^2=ghi,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^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6723298);