Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500456 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2206 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)(3,6)(4,5)(7,8)(17,18)(19,26)(20,25)(21,27)(22,28)(23,29)(24,30)(31,41)(32,40)(33,39)(34,38)(35,42)(36,43)(37,44)(45,48)(46,47);(2,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,19,23,20,24,33,22,32)(18,25,28,26,30,40,29,39)(21,31,35,34,37,46,36,45)(27,38,43,41,44,48,42,47);(1,3,4,7)(2,6,5,8)(9,11)(10,13)(12,16)(14,15)(17,20,23,33,24,32,22,19)(18,26,28,40,30,39,29,25)(21,34,35,46,37,45,36,31)(27,41,43,48,44,47,42,38);(11,16)(13,15)(17,21,24,37)(18,27,30,44)(19,31,33,46)(20,34,32,45)(22,36,23,35)(25,38,40,48)(26,41,39,47)(28,43,29,42);(10,14)(13,15)(17,22,24,23)(18,28,30,29)(19,32,33,20)(21,35,37,36)(25,26,40,39)(27,42,44,43)(31,34,46,45)(38,47,48,41)Polycyclic presentation:
2^9|b^2=gi,c^2=f,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=ehi,d^c=dhi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6413696);