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 | #45265 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1596 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)(17,18,22,29,23,28,38,27)(19,30,33,42,34,24,39,43)(20,35,37,45,21,26,36,25)(31,41,48,40,32,46,47,44);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,14)(13,16)(17,19,23,34)(18,24,28,30)(20,32,21,31)(22,33,38,39)(25,44,45,40)(26,46,35,41)(27,42,29,43)(36,48,37,47);(1,2,5,4)(3,6,8,7)(9,11)(10,13)(12,16)(14,15)(17,20)(18,25)(19,31)(21,23)(22,36)(24,40)(26,29)(27,35)(28,45)(30,44)(32,34)(33,47)(37,38)(39,48)(41,43)(42,46);(1,2,5,4)(3,7,8,6)(11,16)(13,15)(17,21,23,20)(18,26,28,35)(19,32,34,31)(22,36,38,37)(24,41,30,46)(25,27,45,29)(33,47,39,48)(40,42,44,43);(1,4,5,2)(3,7,8,6)(10,14)(13,15)(17,22,23,38)(18,27,28,29)(19,33,34,39)(20,36,21,37)(24,42,30,43)(25,35,45,26)(31,47,32,48)(40,46,44,41)Polycyclic presentation:
2^9|a^2=fgi,b^2=i,c^2=fi,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,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, 6924303);