Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6884841 |
| 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 | #29606 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1508 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,8)(5,6)(17,18,23,29)(19,30,33,25)(20,27,21,35)(22,28,24,39)(26,37,44,34)(31,41,36,45)(32,43,38,42)(40,48,46,47);(2,6)(4,7)(9,10)(11,15)(12,14)(13,16)(17,19,23,33)(18,25,29,30)(20,31,21,36)(22,38,24,32)(26,40,44,46)(27,45,35,41)(28,43,39,42)(34,48,37,47);(1,3,5,8)(2,7,6,4)(9,11)(10,13)(12,16)(14,15)(17,20)(18,26)(19,31)(21,23)(22,34)(24,37)(25,40)(27,39)(28,35)(29,44)(30,46)(32,47)(33,36)(38,48)(41,43)(42,45);(2,6)(3,8)(11,16)(13,15)(17,21,23,20)(18,27,29,35)(19,31,33,36)(22,34,24,37)(25,41,30,45)(26,28,44,39)(32,48,38,47)(40,42,46,43);(1,4,5,7)(2,8,6,3)(10,14)(13,15)(17,22,23,24)(18,28,29,39)(19,32,33,38)(20,34,21,37)(25,42,30,43)(26,35,44,27)(31,47,36,48)(40,45,46,41)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fi,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6797699);