Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533195 |
| 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 | #23853 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1614 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)(3,8)(4,6)(5,7)(17,18,23,36,22,27,21,26)(19,28,37,39,30,24,31,43)(20,32,33,25,34,40,35,41)(29,44,46,42,45,48,47,38);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,29)(21,31)(22,30)(23,37)(25,38)(26,39)(27,28)(32,48)(33,46)(34,45)(35,47)(36,43)(40,44)(41,42);(10,13)(14,16)(17,20,22,34)(18,25,27,41)(19,29,30,45)(21,33,23,35)(24,38,28,42)(26,40,36,32)(31,46,37,47)(39,44,43,48);(1,4,5,3)(2,6,7,8)(9,11)(10,14)(12,15)(13,16)(19,30)(24,28)(29,45)(31,37)(38,42)(39,43)(44,48)(46,47);(11,15)(14,16)(17,21,22,23)(18,26,27,36)(19,31,30,37)(20,33,34,35)(24,39,28,43)(25,40,41,32)(29,46,45,47)(38,44,42,48)Polycyclic presentation:
2^9|a^2=gi,b^2=fi,c^2=i,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6452696);