Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514020 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #36018 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1743 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,18,23,28)(19,24,31,40)(20,25,35,44)(21,26,22,27)(29,38,33,42)(30,39,36,45)(32,41,47,48)(34,43,37,46);(2,6)(7,8)(9,11,12,16)(10,13,14,15)(17,19)(18,24)(20,32)(21,30)(22,36)(23,31)(25,41)(26,39)(27,45)(28,40)(29,34)(33,37)(35,47)(38,43)(42,46)(44,48);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,19)(18,24)(20,33)(21,36)(22,30)(23,31)(25,42)(26,45)(27,39)(28,40)(29,35)(32,37)(34,47)(38,44)(41,46)(43,48);(1,4,5,3)(2,8,6,7)(10,14)(13,15)(17,20)(18,25)(19,29)(21,34)(22,37)(23,35)(24,38)(26,43)(27,46)(28,44)(30,47)(31,33)(32,36)(39,48)(40,42)(41,45);(2,6)(7,8)(10,14)(13,15)(17,21,23,22)(18,26,28,27)(19,30,31,36)(20,34,35,37)(24,39,40,45)(25,43,44,46)(29,47,33,32)(38,48,42,41)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=fg,d^2=fg,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7476845);