Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6660434 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #34633 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1763 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)(3,5)(4,6)(7,8)(10,13)(11,15)(17,18)(19,24)(20,33)(21,26)(22,37)(23,28)(25,35)(27,30)(29,45)(31,40)(32,43)(34,39)(36,44)(38,42)(41,46)(47,48);(3,7)(5,8)(9,10,12,13)(11,14,15,16)(17,19)(18,24)(20,29)(21,30)(22,32)(23,31)(25,39)(26,27)(28,40)(33,45)(34,35)(36,46)(37,43)(38,47)(41,44)(42,48);(1,3)(2,5)(4,7)(6,8)(10,13)(14,16)(17,20,23,36)(18,25,28,42)(19,29,31,46)(21,34,32,47)(22,38,30,35)(24,39,40,48)(26,41,43,45)(27,44,37,33);(3,7)(5,8)(9,11,12,15)(10,14,13,16)(17,21,23,32)(18,26,28,43)(19,30,31,22)(20,34,36,47)(24,27,40,37)(25,41,42,45)(29,35,46,38)(33,39,44,48);(2,6)(5,8)(19,31)(21,32)(24,40)(26,43)(29,46)(34,47)(39,48)(41,45)Polycyclic presentation:
2^9|b^2=f,c^2=i,d^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,e^b=ei,d^c=dfh,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6568060);