Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7200022 |
| 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 | #36009 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1689 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,27)(19,23,29,36)(20,25,31,39)(21,24,33,26)(28,35,42,47)(30,34,43,37)(32,38,44,40)(41,45,48,46);(2,5)(3,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,23)(20,28)(21,30)(22,29)(24,34)(25,35)(26,37)(27,36)(31,42)(32,41)(33,43)(38,45)(39,47)(40,46)(44,48);(11,15)(13,16)(18,24)(21,33)(23,34)(25,38)(26,27)(30,43)(32,44)(35,45)(36,37)(39,40)(41,48)(46,47);(1,3,4,7)(2,6,5,8)(10,14)(11,15)(17,20,22,31)(18,25,27,39)(19,28,29,42)(21,32,33,44)(23,35,36,47)(24,38,26,40)(30,41,43,48)(34,45,37,46);(2,5)(6,8)(11,15)(13,16)(19,29)(20,31)(23,36)(25,39)(30,43)(32,44)(34,37)(38,40)Polycyclic presentation:
2^9|a^2=i,d^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,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, 7108860);