Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7324500 |
| 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 | #45243 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1478 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)(3,6)(4,5)(7,8)(17,18,22,27)(19,28,29,41)(20,24,34,39)(21,25,30,26)(23,32,37,33)(31,43,44,48)(35,40,36,42)(38,46,47,45);(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(19,29)(20,34)(23,37)(24,39)(28,41)(32,33)(35,36)(40,42);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19,30,32,22,29,21,33)(18,23,25,41,27,37,26,28)(20,31,35,46,34,44,36,45)(24,38,42,48,39,47,40,43);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,20)(18,24)(19,31)(21,36)(22,34)(23,38)(25,42)(26,40)(27,39)(28,43)(29,44)(30,35)(32,46)(33,45)(37,47)(41,48);(2,5)(6,8)(10,14)(13,15)(17,21,22,30)(18,25,27,26)(19,32,29,33)(20,35,34,36)(23,28,37,41)(24,40,39,42)(31,45,44,46)(38,48,47,43)Polycyclic presentation:
2^9|a^2=i,c^2=fgi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,d^c=dhi,e^c=ei,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7237253);