Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682596 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #24455 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1561 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(17,18)(19,28)(20,33)(21,27)(22,26)(23,38)(24,30)(25,36)(29,45)(31,44)(32,40)(34,48)(35,42)(37,43)(39,47)(41,46);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,30)(18,24,26,28)(20,34,35,29)(21,31,23,32)(25,41,43,39)(27,40,38,44)(33,45,42,48)(36,47,37,46);(2,6)(5,7)(10,13)(14,16)(17,20,23,36,22,35,21,37)(18,25,27,33,26,43,38,42)(19,29,32,46,30,34,31,47)(24,39,40,48,28,41,44,45);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(18,26)(19,30)(20,35)(27,38)(31,32)(36,37)(39,41)(45,48);(3,8)(5,7)(10,13)(14,16)(17,21,22,23)(18,27,26,38)(19,31,30,32)(20,36,35,37)(24,40,28,44)(25,42,43,33)(29,46,34,47)(39,45,41,48)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=fgi,d^2=hi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6692063);