Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682658 |
| 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 | C2×#135 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1541 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)(19,29)(20,35)(21,28)(22,27)(23,26)(24,33)(25,37)(30,42)(31,41)(32,34)(36,45)(38,44)(39,43)(40,48)(46,47);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,19)(18,24)(20,36)(21,31)(22,33)(23,34)(25,42)(26,41)(27,29)(28,32)(30,39)(35,40)(37,47)(38,48)(43,46)(44,45);(17,20,23,37,22,38,21,39)(18,25,28,35,27,43,26,44)(19,30,34,36,33,47,31,48)(24,40,32,46,29,45,41,42);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(17,21,22,23)(18,26,27,28)(19,31,33,34)(20,37,38,39)(24,41,29,32)(25,35,43,44)(30,36,47,48)(40,46,45,42);(3,11)(6,9)(10,16)(14,15)(17,18)(19,32)(20,25)(21,26)(22,27)(23,28)(24,31)(29,34)(30,46)(33,41)(35,37)(36,45)(38,43)(39,44)(40,48)(42,47)Polycyclic presentation:
2^9|a^2=fi,c^2=g,d^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,d^c=di,h^c=hi,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6591358);