Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7207830 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1499 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)(3,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,23)(20,31)(21,25)(22,27)(24,28)(26,30)(29,37)(32,47)(33,36)(34,48)(35,41)(38,43)(39,44)(40,45)(42,46);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,15)(14,16)(17,19,22,30)(18,23,27,26)(20,28,33,44)(21,29,35,46)(24,36,39,31)(25,37,41,42)(32,43,34,45)(38,48,40,47);(2,6)(5,7)(11,15)(13,16)(17,20,30,44,22,33,19,28)(18,24,23,36,27,39,26,31)(21,34,46,43,35,32,29,45)(25,40,37,47,41,38,42,48);(3,8)(5,7)(10,14)(11,15)(17,21)(18,25)(19,29)(20,32)(22,35)(23,37)(24,38)(26,42)(27,41)(28,43)(30,46)(31,47)(33,34)(36,48)(39,40)(44,45);(2,6)(5,7)(11,15)(13,16)(21,35)(25,41)(29,46)(32,34)(37,42)(38,40)(43,45)(47,48)Polycyclic presentation:
2^9|b^2=i,c^2=fgi,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,d^c=di,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7115594);