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 | #31389 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1408 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(6,8)(9,10)(11,13)(12,14)(15,16)(17,18,23,27,22,28,21,29)(19,30,39,43,33,24,32,42)(20,34,36,25,37,26,38,45)(31,41,47,44,35,46,48,40);(1,2,4,5)(3,6,7,8)(9,10)(11,15)(12,14)(13,16)(17,19,22,33)(18,24,28,30)(20,35,37,31)(21,32,23,39)(25,44,45,40)(26,46,34,41)(27,42,29,43)(36,48,38,47);(9,11)(10,13)(12,16)(14,15)(17,20,22,37)(18,25,28,45)(19,31,33,35)(21,36,23,38)(24,40,30,44)(26,27,34,29)(32,47,39,48)(41,42,46,43);(2,5)(6,8)(11,16)(13,15)(17,20)(18,26)(19,31)(21,38)(22,37)(23,36)(24,41)(25,29)(27,45)(28,34)(30,46)(32,48)(33,35)(39,47)(40,43)(42,44);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,21,22,23)(18,27,28,29)(19,32,33,39)(20,36,37,38)(24,42,30,43)(25,34,45,26)(31,47,35,48)(40,46,44,41)Polycyclic presentation:
2^9|a^2=gi,b^2=f,c^2=i,e^2=f,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, 6918919);