Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667980 |
| 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 | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1577 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)(3,7)(4,6)(5,8)(17,18,22,27,23,28,21,26)(19,24,31,40,30,39,36,43)(20,32,34,42,35,44,37,25)(29,45,47,41,33,48,46,38);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(17,19)(18,24)(20,33)(21,36)(22,31)(23,30)(25,41)(26,43)(27,40)(28,39)(29,35)(32,48)(34,46)(37,47)(38,42)(44,45);(3,8)(5,7)(9,11,12,16)(10,13,14,15)(17,20)(18,25)(19,29)(21,34)(22,37)(23,35)(24,38)(26,32)(27,44)(28,42)(30,33)(31,46)(36,47)(39,41)(40,48)(43,45);(2,6)(5,7)(19,30)(24,39)(29,33)(31,36)(38,41)(40,43)(45,48)(46,47);(10,14)(13,15)(17,21,23,22)(18,26,28,27)(19,31,30,36)(20,34,35,37)(24,40,39,43)(25,32,42,44)(29,46,33,47)(38,48,41,45)Polycyclic presentation:
2^9|a^2=g,c^2=fhi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,g^c=gi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6664997);