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 | #42796 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1763 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,13)(12,14)(15,16)(17,18)(19,29)(20,26)(21,28)(22,24)(23,27)(25,33)(30,40)(31,43)(32,37)(34,38)(35,42)(36,41)(39,45)(44,48)(46,47);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(18,24)(19,30)(20,34)(27,28)(32,33)(35,36)(39,47)(43,48);(9,11)(10,13)(12,16)(14,15)(17,19,22,30)(18,25,24,37)(20,31,34,44)(21,33,23,32)(26,39,38,47)(27,29,28,40)(35,46,36,45)(41,48,42,43);(3,8)(5,7)(11,16)(13,15)(17,20)(18,26)(19,31)(21,36)(22,34)(23,35)(24,38)(25,39)(27,42)(28,41)(29,43)(30,44)(32,46)(33,45)(37,47)(40,48);(10,14)(13,15)(17,21,22,23)(18,27,24,28)(19,32,30,33)(20,35,34,36)(25,40,37,29)(26,41,38,42)(31,45,44,46)(39,43,47,48)Polycyclic presentation:
2^9|a^2=fi,c^2=i,e^2=i,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, 6902124);