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 | #3045 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1560 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)(17,18,24,27)(19,25,32,28)(20,23,35,21)(22,26,39,44)(29,33,31,41)(30,40,47,45)(34,38,43,36)(37,42,46,48);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,19)(18,25)(20,33)(21,29)(22,37)(23,31)(24,32)(26,42)(27,28)(30,38)(34,45)(35,41)(36,47)(39,46)(40,43)(44,48);(17,20,23,27,24,35,21,18)(19,28,31,41,32,25,29,33)(22,34,38,44,39,43,36,26)(30,45,46,48,47,40,37,42);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(17,21,24,23)(18,20,27,35)(19,29,32,31)(22,36,39,38)(25,33,28,41)(26,34,44,43)(30,37,47,46)(40,42,45,48);(3,11)(6,9)(10,16)(14,15)(17,22)(18,26)(19,30)(20,34)(21,36)(23,38)(24,39)(25,40)(27,44)(28,45)(29,37)(31,46)(32,47)(33,42)(35,43)(41,48)Polycyclic presentation:
2^9|a^2=fi,c^2=g,d^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,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, 6591362);