Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6807031 |
| 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 | |
1620 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,9,11,6)(4,12,13,7)(10,14,16,15)(17,18)(19,25)(20,35)(21,27)(22,28)(23,29)(24,30)(26,36)(31,46)(32,42)(33,43)(34,45)(37,47)(38,48)(39,41)(40,44);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,19)(18,25)(20,22)(21,32)(23,40)(24,34)(26,46)(27,42)(28,35)(29,44)(30,45)(31,36)(33,38)(37,39)(41,47)(43,48);(17,20,23,36,24,37,21,33)(18,26,29,47,30,43,27,35)(19,31,32,22,34,38,40,39)(25,41,42,46,45,28,44,48);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(17,21,24,23)(18,27,30,29)(19,32,34,40)(20,33,37,36)(22,38,39,31)(25,42,45,44)(26,35,43,47)(28,48,41,46);(3,11)(6,9)(10,16)(14,15)(17,22,23,38,24,39,21,31)(18,28,29,48,30,41,27,46)(19,33,32,20,34,36,40,37)(25,43,42,35,45,26,44,47)Polycyclic presentation:
2^9|a^2=f,c^2=h,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,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6716301);