Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1572 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,27,22,26,35,24)(19,23,30,36,28,38,44,39)(20,31,34,40,33,25,41,43)(29,42,45,48,32,37,46,47);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,19,22,28)(18,23,26,38)(20,32,33,29)(21,30,35,44)(24,39,27,36)(25,42,31,37)(34,46,41,45)(40,47,43,48);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,13,14,15)(18,24)(19,28)(20,33)(21,35)(23,36)(25,43)(26,27)(31,40)(37,48)(38,39)(42,47)(45,46);(1,3,5,4)(2,8,6,7)(11,16)(13,15)(17,20,22,33)(18,25,26,31)(19,29,28,32)(21,34,35,41)(23,37,38,42)(24,40,27,43)(30,45,44,46)(36,47,39,48);(1,4,5,3)(2,8,6,7)(10,14)(13,15)(18,26)(19,28)(20,33)(24,27)(30,44)(34,41)(37,42)(47,48)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=fhi,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6982105);