Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 | |
1923 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,18)(19,24)(20,30)(21,26)(22,37)(23,27)(25,31)(28,41)(29,38)(32,46)(33,39)(34,47)(35,43)(36,48)(40,45)(42,44);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,14)(13,16)(17,19,23,22)(18,24,27,37)(20,31,33,28)(21,29,35,36)(25,39,41,30)(26,38,43,48)(32,45,34,44)(40,47,42,46);(1,4,5,2)(3,7,8,6)(9,11)(10,13)(12,16)(14,15)(17,20,22,31,23,33,19,28)(18,25,24,30,27,41,37,39)(21,34,36,44,35,32,29,45)(26,42,38,47,43,40,48,46);(3,8)(6,7)(11,16)(13,15)(17,21)(18,26)(19,29)(20,32)(22,36)(23,35)(24,38)(25,40)(27,43)(28,44)(30,46)(31,45)(33,34)(37,48)(39,47)(41,42);(10,14)(13,15)(18,27)(20,33)(21,35)(24,37)(28,31)(29,36)(40,42)(46,47)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fg,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^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6927243);