Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ 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:
(2,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,26)(19,28,31,23)(20,25,33,43)(22,36,35,27)(24,37,41,34)(29,39,30,42)(32,40,48,45)(38,46,44,47);(1,2)(3,8)(4,7)(5,6)(9,10)(11,15)(12,14)(13,16)(17,19)(18,23)(20,29)(21,31)(22,32)(24,38)(25,42)(26,28)(27,40)(30,33)(34,46)(35,48)(36,45)(37,47)(39,43)(41,44);(9,11)(10,13)(12,16)(14,15)(17,20,21,33)(18,24,26,41)(19,29,31,30)(22,37,35,34)(23,38,28,44)(25,27,43,36)(32,47,48,46)(39,45,42,40);(1,3,5,4)(2,7,6,8)(11,16)(13,15)(17,20)(18,25)(19,30)(21,33)(22,34)(23,39)(24,36)(26,43)(27,41)(28,42)(29,31)(32,47)(35,37)(38,40)(44,45)(46,48);(1,4,5,3)(2,7,6,8)(10,14)(13,15)(19,31)(20,33)(23,28)(24,41)(25,43)(32,48)(34,37)(40,45)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=fi,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,e^c=ei,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, 6908194);