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 | #42679 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1695 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)(11,15)(14,16)(17,18,21,26)(19,28,29,23)(20,25,32,40)(22,24,34,27)(30,41,42,36)(31,35,43,37)(33,38,45,39)(44,46,48,47);(2,5)(3,7)(10,13)(11,15)(17,19)(18,23)(20,30)(21,29)(22,31)(24,37)(25,36)(26,28)(27,35)(32,42)(33,44)(34,43)(38,47)(39,46)(40,41)(45,48);(9,10)(11,14)(12,13)(15,16)(18,24)(19,29)(20,32)(22,34)(23,35)(25,39)(26,27)(28,37)(36,47)(38,40)(41,46)(44,48);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(17,20,21,32)(18,25,26,40)(19,30,29,42)(22,33,34,45)(23,36,28,41)(24,38,27,39)(31,44,43,48)(35,46,37,47);(9,11,12,15)(10,14,13,16)(19,29)(20,32)(23,28)(25,40)(31,43)(33,45)(35,37)(38,39)Polycyclic presentation:
2^9|a^2=i,d^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ehi,g^a=gi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,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, 7052740);