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 |
| prime radical | #45316 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1754 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)(17,18,21,23)(19,27,30,40)(20,25,32,36)(22,34,28,26)(24,37,35,31)(29,41,43,48)(33,45,42,39)(38,47,46,44);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,14)(13,16)(18,23)(20,32)(24,35)(26,34)(27,40)(29,43)(33,42)(44,47);(1,2,5,4)(3,6,8,7)(9,11)(10,13)(12,16)(14,15)(17,19,28,37,21,30,22,31)(18,24,34,40,23,35,26,27)(20,29,42,47,32,43,33,44)(25,38,45,48,36,46,39,41);(1,2,5,4)(3,7,8,6)(11,16)(13,15)(17,20)(18,25)(19,29)(21,32)(22,33)(23,36)(24,38)(26,39)(27,41)(28,42)(30,43)(31,44)(34,45)(35,46)(37,47)(40,48);(1,4,5,2)(3,7,8,6)(10,14)(13,15)(19,30)(20,32)(24,35)(25,36)(27,40)(31,37)(33,42)(39,45)Polycyclic presentation:
2^9|a^2=f,c^2=fg,d^2=fi,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,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, 6938926);