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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1527 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,18,21,27,22,26,34,24)(19,23,30,35,28,37,43,38)(20,31,33,39,32,25,40,41)(29,44,45,48,42,36,47,46);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,14)(13,16)(17,19)(18,23)(20,29)(21,30)(22,28)(24,38)(25,36)(26,37)(27,35)(31,44)(32,42)(33,45)(34,43)(39,48)(40,47)(41,46);(9,11,12,16)(10,13,14,15)(18,24)(19,28)(20,32)(21,34)(23,35)(25,41)(26,27)(31,39)(36,48)(37,38)(44,46)(45,47);(1,2,5,4)(3,7,8,6)(11,16)(13,15)(17,20,22,32)(18,25,26,31)(19,29,28,42)(21,33,34,40)(23,36,37,44)(24,39,27,41)(30,45,43,47)(35,46,38,48);(1,4,5,2)(3,7,8,6)(10,14)(13,15)(18,26)(19,28)(20,32)(24,27)(30,43)(33,40)(36,44)(46,48)Polycyclic presentation:
2^9|a^2=fg,c^2=hi,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,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, 6980433);