Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667900 |
| 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 | #2407 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1487 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)(17,18,22,28)(19,29,30,24)(20,34,36,26)(21,27,23,25)(31,39,44,43)(32,40,33,38)(35,42,37,41)(45,48,46,47);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,19)(18,24)(20,31)(21,32)(22,30)(23,33)(25,40)(26,39)(27,38)(28,29)(34,43)(35,45)(36,44)(37,46)(41,48)(42,47);(10,14)(15,16)(18,25)(19,30)(21,23)(24,38)(26,41)(27,28)(29,40)(31,44)(34,42)(35,37)(39,47)(43,48);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,20)(18,26)(19,31)(21,37)(22,36)(23,35)(24,39)(25,41)(27,42)(28,34)(29,43)(30,44)(32,46)(33,45)(38,47)(40,48);(3,8)(5,7)(9,12,13,11)(10,15,14,16)(17,21,22,23)(18,27,28,25)(19,32,30,33)(20,35,36,37)(24,38,29,40)(26,41,34,42)(31,45,44,46)(39,48,43,47)Polycyclic presentation:
2^9|a^2=i,d^2=hi,e^2=h,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, 6765943);