Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907869 |
| 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 | #38095 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1621 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(9,10,12,13)(11,15,16,14)(17,18)(19,28)(20,33)(21,26)(22,25)(23,30)(24,36)(27,34)(29,43)(31,41)(32,38)(35,39)(37,44)(40,48)(42,46)(45,47);(1,2)(3,7)(4,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,29)(21,32)(22,30)(24,37)(25,28)(26,41)(27,38)(31,34)(33,47)(35,45)(36,46)(39,43)(40,42)(44,48);(1,3)(2,5)(4,8)(6,7)(10,13)(14,15)(17,20,34,48,22,35,21,36)(18,24,26,39,25,40,27,33)(19,29,31,44,30,45,32,46)(23,37,41,43,28,42,38,47);(3,8)(5,7)(9,11,12,16)(10,14,13,15)(18,25)(19,30)(20,35)(26,27)(31,32)(36,48)(37,42)(43,47);(2,6)(5,7)(17,21,22,34)(18,26,25,27)(19,31,30,32)(20,36,35,48)(23,38,28,41)(24,39,40,33)(29,44,45,46)(37,47,42,43)Polycyclic presentation:
2^9|a^2=fi,c^2=fg,d^2=fi,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=dfh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6818192);