Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470185 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1577 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)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18,24,29)(19,25,21,27)(20,32,35,43)(22,28,23,37)(26,38,44,34)(30,45,33,48)(31,40,36,41)(39,46,42,47);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,25)(20,30)(21,24)(22,36)(23,31)(26,39)(27,29)(28,41)(32,45)(33,35)(34,47)(37,40)(38,46)(42,44)(43,48);(1,3,5,4)(2,7,8,6)(11,15)(13,16)(17,20)(18,26)(19,30)(21,33)(22,34)(23,38)(24,35)(25,39)(27,42)(28,43)(29,44)(31,46)(32,37)(36,47)(40,45)(41,48);(1,4,5,3)(2,6,8,7)(11,15)(13,16)(17,21,24,19)(18,27,29,25)(20,33,35,30)(22,31,23,36)(26,42,44,39)(28,40,37,41)(32,48,43,45)(34,46,38,47);(10,14)(13,16)(17,22,24,23)(18,28,29,37)(19,31,21,36)(20,34,35,38)(25,40,27,41)(26,43,44,32)(30,46,33,47)(39,45,42,48)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=f,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dhi,e^b=ei,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6378935);