Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499823 |
| 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 | #2414 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1592 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)(17,18,22,27)(19,23,29,36)(20,25,31,39)(21,24,33,26)(28,35,42,47)(30,34,43,37)(32,38,44,40)(41,45,48,46);(2,6)(7,8)(9,10)(11,15)(12,14)(13,16)(17,19)(18,23)(20,28)(21,30)(22,29)(24,34)(25,35)(26,37)(27,36)(31,42)(32,41)(33,43)(38,45)(39,47)(40,46)(44,48);(1,3,5,4)(2,7,6,8)(9,11)(10,13)(12,16)(14,15)(18,24)(21,33)(23,34)(25,38)(26,27)(30,43)(32,44)(35,45)(36,37)(39,40)(41,48)(46,47);(1,4,5,3)(2,8,6,7)(11,16)(13,15)(17,20,22,31)(18,25,27,39)(19,28,29,42)(21,32,33,44)(23,35,36,47)(24,38,26,40)(30,41,43,48)(34,45,37,46);(10,14)(13,15)(19,29)(20,31)(23,36)(25,39)(30,43)(32,44)(34,37)(38,40)Polycyclic presentation:
2^9|a^2=i,c^2=f,d^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=ehi,d^c=dh,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6408789);