Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61629 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | #61234 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#436 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1633 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)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,22,27,23,25,21,35)(19,28,31,24,29,39,36,37)(20,26,34,41,32,40,33,45)(30,42,44,38,43,48,46,47);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(19,29)(20,32)(24,37)(26,40)(28,39)(31,36)(33,34)(41,45);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,19)(18,24)(20,30)(21,36)(22,31)(23,29)(25,37)(26,38)(27,39)(28,35)(32,43)(33,46)(34,44)(40,47)(41,48)(42,45);(3,11)(6,9)(10,16)(13,15)(18,25)(20,32)(24,37)(27,35)(28,39)(30,43)(33,34)(44,46);(2,8)(6,9)(7,14)(13,15)(17,20,22,34,23,32,21,33)(18,26,27,41,25,40,35,45)(19,30,31,44,29,43,36,46)(24,38,39,48,37,47,28,42)Polycyclic presentation:
2^9|a^2=h,c^2=f,e^2=h,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,g^a=gi,c^b=ci,d^b=df,e^b=ei,g^c=gi,e^d=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6306276);