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 |
| prime radical | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1291 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,18,23,25,22,28,21,27)(19,24,34,36,30,38,31,39)(20,26,35,40,32,41,33,42)(29,37,45,46,44,48,43,47);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,14)(13,16)(17,19)(18,24)(20,29)(21,31)(22,30)(23,34)(25,36)(26,37)(27,39)(28,38)(32,44)(33,43)(35,45)(40,46)(41,48)(42,47);(1,2,5,4)(3,6,8,7)(9,11)(10,13)(12,16)(14,15)(18,25)(21,23)(24,36)(26,40)(27,28)(31,34)(33,35)(37,46)(38,39)(41,42)(43,45)(47,48);(1,4,5,2)(3,6,8,7)(11,16)(13,15)(17,20,22,32)(18,26,28,41)(19,29,30,44)(21,33,23,35)(24,37,38,48)(25,40,27,42)(31,43,34,45)(36,46,39,47);(10,14)(13,15)(19,30)(20,32)(24,38)(26,41)(31,34)(33,35)(36,39)(40,42)Polycyclic presentation:
2^9|a^2=fgi,c^2=f,d^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=ehi,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, 6916101);