Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573591 |
| 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 | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1473 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,6)(4,8)(5,7)(17,18,23,35,22,27,21,26)(19,28,30,24,32,38,31,39)(20,25,36,46,33,40,34,41)(29,42,44,37,43,48,45,47);(1,3,5,4)(2,6,7,8)(9,10,12,13)(11,14,15,16)(17,19)(18,24)(20,29)(21,30)(22,32)(23,31)(25,37)(26,38)(27,39)(28,35)(33,43)(34,44)(36,45)(40,47)(41,48)(42,46);(2,7)(6,8)(10,13)(11,15)(17,20)(18,25)(19,29)(21,34)(22,33)(23,36)(24,37)(26,41)(27,40)(28,42)(30,44)(31,45)(32,43)(35,46)(38,48)(39,47);(1,4,5,3)(2,8,7,6)(9,11)(10,14)(12,15)(13,16)(20,33)(25,40)(29,43)(34,36)(37,47)(41,46)(42,48)(44,45);(10,13)(11,15)(17,21,22,23)(18,26,27,35)(19,30,32,31)(20,34,33,36)(24,38,39,28)(25,41,40,46)(29,44,43,45)(37,48,47,42)Polycyclic presentation:
2^9|a^2=fi,b^2=gh,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^d=eh83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6520241);