Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996355 |
| 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 | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1680 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)(6,7)(9,10)(11,13)(12,14)(15,16)(17,18,23,28)(19,29,30,40)(20,25,36,43)(21,27,22,26)(24,32,38,33)(31,45,34,48)(35,44,37,42)(39,47,41,46);(1,2,4,5)(3,7,8,6)(9,10)(11,15)(12,14)(13,16)(19,30)(24,38)(29,40)(31,34)(32,33)(39,41)(45,48)(46,47);(9,11)(10,13)(12,16)(14,15)(17,19,21,32,23,30,22,33)(18,24,26,40,28,38,27,29)(20,34,37,46,36,31,35,47)(25,41,44,45,43,39,42,48);(1,3,4,8)(2,6,5,7)(11,16)(13,15)(17,20)(18,25)(19,31)(21,37)(22,35)(23,36)(24,39)(26,44)(27,42)(28,43)(29,45)(30,34)(32,47)(33,46)(38,41)(40,48);(10,14)(13,15)(17,21,23,22)(18,26,28,27)(19,32,30,33)(20,35,36,37)(24,40,38,29)(25,42,43,44)(31,46,34,47)(39,45,41,48)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=gi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6929280);