Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667900 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1773 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)(3,7)(4,6)(5,8)(17,18,22,24)(19,28,29,42)(20,33,35,26)(21,27,23,37)(25,31,38,32)(30,45,44,43)(34,41,36,40)(39,46,48,47);(1,3,4,8)(2,5,6,7)(9,10)(11,14)(12,13)(15,16)(18,24)(19,29)(26,33)(27,37)(30,44)(31,32)(40,41)(46,47);(10,13)(14,16)(17,19,22,29)(18,25,24,38)(20,30,35,44)(21,32,23,31)(26,39,33,48)(27,42,37,28)(34,47,36,46)(40,43,41,45);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,20)(18,26)(19,30)(21,36)(22,35)(23,34)(24,33)(25,39)(27,41)(28,43)(29,44)(31,47)(32,46)(37,40)(38,48)(42,45);(3,8)(5,7)(10,13)(14,16)(17,21,22,23)(18,27,24,37)(19,31,29,32)(20,34,35,36)(25,28,38,42)(26,40,33,41)(30,46,44,47)(39,45,48,43)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=i,d^2=hi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,e^c=ei,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, 6674815);