Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6353061 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2878 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,6,10)(3,11,13,8)(4,9,5,15)(7,14,16,12)(17,18,22,26)(19,23,29,35)(20,24,31,37)(21,25,32,38)(27,39,41,33)(28,34,42,45)(30,36,43,46)(40,47,48,44);(1,3)(2,7)(4,12)(5,14)(6,13)(8,9)(10,16)(11,15)(17,19)(18,23)(20,27)(21,28)(22,29)(24,33)(25,34)(26,35)(30,40)(31,41)(32,42)(36,44)(37,39)(38,45)(43,48)(46,47);(17,20,22,31)(18,24,26,37)(19,27,29,41)(21,30,32,43)(23,33,35,39)(25,36,38,46)(28,40,42,48)(34,44,45,47);(1,3)(2,8)(4,14)(5,12)(6,13)(7,9)(10,11)(15,16)(17,21)(18,25)(19,28)(20,30)(22,32)(23,34)(24,36)(26,38)(27,40)(29,42)(31,43)(33,44)(35,45)(37,46)(39,47)(41,48);(1,4,6,5)(2,9,10,15)(3,12,13,14)(7,8,16,11)(21,32)(25,38)(28,42)(30,43)(34,45)(36,46)(40,48)(44,47)Polycyclic presentation:
2^9|a^2=hi,c^2=i,e^2=h,f^2=h,b^a=bf,c^a=cg,d^a=dh,f^a=fh,f^b=fh,g^b=gi,f^c=fi,e^d=ehi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6260469);