Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6814815 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1668 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)(23,30)(24,28)(25,34)(31,45)(33,47)(35,40)(36,48)(37,42)(38,46)(39,44)(41,43);(1,3)(2,7)(4,11)(5,12)(6,13)(8,15)(9,10)(14,16)(17,19)(18,20)(21,36)(22,32)(23,25)(24,40)(26,29)(27,43)(28,34)(30,35)(31,38)(33,46)(37,41)(39,47)(42,48)(44,45);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,15,10,16)(17,20,22,29)(18,25)(19,30)(21,33,37,45)(23,34,24,35)(26,40)(27,41)(28,32)(31,46)(36,42)(38,43,39,48)(44,47);(1,5,6,4)(2,8,9,14)(3,12,13,11)(7,15,10,16)(18,26)(20,29)(21,37)(28,30)(31,47)(34,35)(36,41)(38,39);(3,13)(7,10)(11,12)(15,16)(17,21,24,39,22,37,23,38)(18,27,28,44,26,42,30,46)(19,31,25,41,32,47,40,36)(20,33,35,48,29,45,34,43)Polycyclic presentation:
2^9|c^2=fgh,d^2=fi,e^2=h,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,d^c=di,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6722953);