Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| 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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1530 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,25,21,27,22,28)(19,29,37,41,31,24,32,39)(20,33,38,44,35,26,36,42)(30,40,46,47,34,43,45,48);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,19,21,31)(18,24,27,29)(20,34,35,30)(22,32,23,37)(25,39,28,41)(26,43,33,40)(36,46,38,45)(42,48,44,47);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,13,14,15)(18,25)(22,23)(24,39)(26,42)(27,28)(29,41)(32,37)(33,44)(36,38)(40,47)(43,48)(45,46);(1,3,5,4)(2,8,6,7)(11,16)(13,15)(17,20,21,35)(18,26,27,33)(19,30,31,34)(22,36,23,38)(24,40,29,43)(25,42,28,44)(32,45,37,46)(39,47,41,48);(1,4,5,3)(2,8,6,7)(10,14)(13,15)(19,31)(20,35)(24,29)(26,33)(32,37)(36,38)(39,41)(42,44)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=fhi,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=ehi,d^c=dh,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6982056);