Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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 | #45412 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1589 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)(17,18,24,28,23,30,22,29)(19,25,33,46,35,42,34,41)(20,36,37,26,21,27,38,44)(31,45,48,39,32,40,47,43);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,14)(13,16)(17,19)(18,25)(20,32)(21,31)(22,34)(23,35)(24,33)(26,43)(27,45)(28,46)(29,41)(30,42)(36,40)(37,47)(38,48)(39,44);(1,2,5,4)(3,6,8,7)(9,11)(10,13)(12,16)(14,15)(17,20)(18,26)(19,31)(21,23)(22,37)(24,38)(25,39)(27,29)(28,36)(30,44)(32,35)(33,47)(34,48)(40,41)(42,43)(45,46);(1,2,5,4)(3,7,8,6)(11,16)(13,15)(17,21,23,20)(18,27,30,36)(19,32,35,31)(22,37,24,38)(25,40,42,45)(26,28,44,29)(33,47,34,48)(39,46,43,41);(1,4,5,2)(3,7,8,6)(10,14)(13,15)(17,22,23,24)(18,28,30,29)(19,33,35,34)(20,37,21,38)(25,41,42,46)(26,36,44,27)(31,47,32,48)(39,40,43,45)Polycyclic presentation:
2^9|a^2=fgi,c^2=f,d^2=fi,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6924358);