Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667964 |
| 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 | #3005 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1631 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)(3,9)(4,8)(5,7)(6,11)(10,16)(12,14)(13,15)(17,18)(19,25)(20,34)(21,27)(22,28)(23,29)(24,30)(26,37)(31,47)(32,42)(33,44)(35,41)(36,45)(38,48)(39,46)(40,43);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,19,24,33)(18,25,30,44)(20,35,38,39)(21,32,23,40)(22,36,31,37)(26,28,45,47)(27,42,29,43)(34,41,48,46);(17,20)(18,26)(19,31)(21,36)(22,33)(23,37)(24,38)(25,41)(27,34)(28,43)(29,48)(30,45)(32,39)(35,40)(42,47)(44,46);(2,7)(6,9)(8,15)(14,16)(17,21,24,23)(18,27,30,29)(19,32,33,40)(20,36,38,37)(22,35,31,39)(25,42,44,43)(26,34,45,48)(28,41,47,46);(1,4,5,13)(2,8,7,15)(3,10,11,12)(6,14,9,16)(17,22,24,31)(18,28,30,47)(19,20,33,38)(21,39,23,35)(25,34,44,48)(26,43,45,42)(27,46,29,41)(32,37,40,36)Polycyclic presentation:
2^9|b^2=fi,d^2=i,e^2=fi,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,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, 6581003);