Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682638 |
| 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 | #60937 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1643 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
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)(17,18,23,28)(19,24,29,39)(20,32,31,44)(21,25,38,43)(22,27,33,26)(30,40,46,48)(34,47,35,41)(36,45,37,42);(1,3,5,11)(2,6,7,9)(4,12,13,10)(8,16,15,14)(17,19,23,29)(18,24,28,39)(20,22,31,33)(21,35,38,34)(25,41,43,47)(26,32,27,44)(30,36,46,37)(40,45,48,42);(17,20,33,29,23,31,22,19)(18,24,27,44,28,39,26,32)(21,34,37,46,38,35,36,30)(25,40,45,41,43,48,42,47);(2,7)(6,9)(8,15)(14,16)(19,29)(20,31)(24,39)(30,46)(32,44)(34,35)(40,48)(41,47);(1,4,5,13)(2,8,7,15)(3,10,11,12)(6,14,9,16)(17,21)(18,25)(19,30)(20,34)(22,36)(23,38)(24,40)(26,42)(27,45)(28,43)(29,46)(31,35)(32,47)(33,37)(39,48)(41,44)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=gi,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,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, 6593323);