Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6342800 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×C22⋊C4 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C24 | C25 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#124 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
3064 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18)(19,23)(20,30,32,24)(21,25)(22,26)(27,41,43,35)(28,36)(29,37)(31,44,46,38)(33,39)(34,40)(42,48,45,47);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19,22,29)(18,23,26,37)(20,27,32,43)(21,33,34,28)(24,35,30,41)(25,39,40,36)(31,45,46,42)(38,48,44,47);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20)(18,24)(19,27)(21,31)(22,32)(23,35)(25,38)(26,30)(28,42)(29,43)(33,45)(34,46)(36,47)(37,41)(39,48)(40,44);(3,11)(6,9)(10,16)(13,15)(21,34)(25,40)(28,33)(31,46)(36,39)(38,44)(42,45)(47,48);(2,8)(6,9)(7,14)(13,15)(17,21,22,34)(18,25,26,40)(19,28,29,33)(20,31,32,46)(23,36,37,39)(24,38,30,44)(27,42,43,45)(35,47,41,48)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=f,e^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,e^b=ei,h^c=hi,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6293313);