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 |
| prime radical | #61223 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1672 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)(17,18,22,27)(19,30,32,25)(20,33,35,46)(21,28,40,41)(23,39,24,29)(26,38,43,37)(31,36,45,48)(34,44,47,42);(1,3)(2,7)(4,11)(5,12)(6,13)(8,15)(9,10)(14,16)(17,19)(18,25)(20,34)(21,39)(22,32)(23,28)(24,41)(26,36)(27,30)(29,40)(31,38)(33,42)(35,47)(37,45)(43,48)(44,46);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,15,10,16)(17,20,22,35)(18,26)(19,31)(21,36,40,48)(23,37,24,38)(25,42,30,44)(27,43)(28,34)(29,33)(32,45)(39,46)(41,47);(1,5,6,4)(2,8,9,14)(3,12,13,11)(7,15,10,16)(18,27)(20,35)(21,40)(25,30)(29,39)(31,45)(34,47)(37,38);(3,13)(7,10)(11,12)(15,16)(17,21,24,30,22,40,23,25)(18,28,29,32,27,41,39,19)(20,36,38,44,35,48,37,42)(26,34,33,45,43,47,46,31)Polycyclic presentation:
2^9|a^2=i,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, 6722955);