Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667748 |
| 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 | #3488 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1691 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)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18)(19,24)(20,26)(21,25)(22,27)(23,28)(29,38)(30,37)(31,39)(32,40)(33,42)(34,41)(35,44)(36,43)(45,48)(46,47);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,19,23,32)(18,24,28,40)(20,33,35,45)(21,31,22,29)(25,39,27,38)(26,42,44,48)(30,36,46,34)(37,43,47,41);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(18,25)(19,29)(21,22)(26,41)(27,28)(30,33)(31,32)(34,36)(38,39)(42,48)(43,44)(45,46);(2,8)(6,9)(7,14)(13,15);(3,11)(6,9)(10,16)(13,15)(17,20)(18,26)(19,30)(21,34)(22,36)(23,35)(24,37)(25,41)(27,43)(28,44)(29,33)(31,45)(32,46)(38,42)(39,48)(40,47)Polycyclic presentation:
2^9|b^2=fi,c^2=f,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,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6576894);