Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907993 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #23923 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1672 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,6)(3,7,8,4)(17,18,22,27)(19,28,23,24)(20,31,34,38)(21,35,36,26)(25,29,41,32)(30,39,37,42)(33,46,47,44)(40,45,48,43);(2,6)(4,7)(9,10)(11,15)(12,14)(13,16)(17,19,22,23)(18,24,27,28)(20,32,34,29)(21,30,36,37)(25,31,41,38)(26,39,35,42)(33,45,47,43)(40,44,48,46);(1,3)(2,7)(4,6)(5,8)(9,11)(10,13)(12,16)(14,15)(17,20,19,29,22,34,23,32)(18,25,24,38,27,41,28,31)(21,33,30,43,36,47,37,45)(26,40,39,46,35,48,42,44);(2,6)(3,8)(11,16)(13,15)(17,21)(18,26)(19,30)(20,33)(22,36)(23,37)(24,39)(25,40)(27,35)(28,42)(29,43)(31,44)(32,45)(34,47)(38,46)(41,48);(1,4,5,7)(2,3,6,8)(10,14)(13,15)(21,36)(26,35)(30,37)(33,47)(39,42)(40,48)(43,45)(44,46)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=gi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=dfh,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6819538);