Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682638 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #61224 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1627 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)(3,10)(4,8)(5,14)(6,9)(7,13)(11,16)(12,15)(17,18)(19,29)(20,32)(21,27)(22,26)(23,37)(24,28)(25,31)(30,45)(33,48)(34,40)(35,41)(36,44)(38,43)(39,46)(42,47);(1,3)(2,7)(4,11)(5,12)(6,13)(8,15)(9,10)(14,16)(17,19)(18,25)(20,23)(21,33)(22,31)(24,34)(26,29)(27,42)(28,32)(30,38)(35,47)(36,46)(37,40)(39,43)(41,48)(44,45);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,15,10,16)(17,20,24,19,22,34,23,31)(18,25,37,40,26,29,28,32)(21,33,38,30,35,47,36,46)(27,39,44,42,41,45,43,48);(1,5,6,4)(2,8,9,14)(3,12,13,11)(7,15,10,16)(18,26)(21,35)(25,29)(28,37)(30,46)(32,40)(33,47)(36,38);(3,13)(7,10)(11,12)(15,16)(17,21,22,35)(18,27,26,41)(19,30,31,46)(20,33,34,47)(23,36,24,38)(25,39,29,45)(28,43,37,44)(32,48,40,42)Polycyclic presentation:
2^9|c^2=fg,d^2=fi,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6591757);