Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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 | #2388 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1708 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(6,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,28)(19,29,33,24)(20,26,36,42)(21,27,23,38)(25,37,43,35)(30,40,31,44)(32,46,34,41)(39,47,45,48);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16)(17,19)(18,24)(20,30)(21,32)(22,33)(23,34)(25,39)(26,44)(27,41)(28,29)(31,36)(35,47)(37,48)(38,46)(40,42)(43,45);(1,3,4,7)(2,6,5,8)(9,11)(10,13)(12,16)(14,15)(17,20,22,36)(18,25,28,43)(19,30,33,31)(21,35,23,37)(24,39,29,45)(26,27,42,38)(32,47,34,48)(40,46,44,41);(2,5)(6,8)(11,16)(13,15)(17,20)(18,26)(19,31)(21,37)(22,36)(23,35)(24,40)(25,38)(27,43)(28,42)(29,44)(30,33)(32,47)(34,48)(39,41)(45,46);(10,14)(13,15)(17,21,22,23)(18,27,28,38)(19,32,33,34)(20,35,36,37)(24,41,29,46)(25,42,43,26)(30,47,31,48)(39,40,45,44)Polycyclic presentation:
2^9|a^2=i,c^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,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, 6902440);