Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667580 |
| 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 | #31066 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1575 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,7)(3,8,4,6)(9,10,12,14)(11,15,16,13)(17,18,23,27)(19,24,31,38)(20,25,32,40)(21,35,22,26)(28,36,30,39)(29,37,44,48)(33,45,34,41)(42,46,43,47);(1,2)(3,6)(4,8)(5,7)(9,11)(10,13)(12,16)(14,15)(17,19)(18,24)(20,29)(21,30)(22,28)(23,31)(25,37)(26,36)(27,38)(32,44)(33,43)(34,42)(35,39)(40,48)(41,46)(45,47);(10,14)(13,15)(19,28)(20,32)(21,22)(24,36)(25,40)(26,35)(29,43)(30,31)(37,47)(38,39)(42,44)(46,48);(1,3,5,4)(2,6,7,8)(10,14)(13,15)(17,20,23,32)(18,25,27,40)(19,29,31,44)(21,33,22,34)(24,37,38,48)(26,41,35,45)(28,42,30,43)(36,46,39,47);(1,4,5,3)(2,6,7,8)(11,16)(13,15)(17,21)(18,26)(20,33)(22,23)(24,38)(25,41)(27,35)(28,30)(32,34)(37,48)(40,45)(42,43)Polycyclic presentation:
2^9|a^2=fi,d^2=fgi,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6649394);