Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996355 |
| 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 | #45385 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1488 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)(11,15)(12,16)(13,14)(17,18,23,27)(19,24,22,37)(20,30,33,46)(21,26,35,42)(25,28,40,32)(29,38,36,48)(31,45,34,47)(39,43,41,44);(1,3)(2,6)(4,7)(5,8)(10,14)(11,12)(17,19,23,22)(18,24,27,37)(20,28,33,32)(21,29,35,36)(25,30,40,46)(26,38,42,48)(31,43,34,44)(39,45,41,47);(9,10)(11,16)(12,15)(13,14)(17,20,19,28,23,33,22,32)(18,25,37,46,27,40,24,30)(21,34,29,44,35,31,36,43)(26,41,48,45,42,39,38,47);(1,2,5,4)(3,7,8,6)(9,11,13,12)(10,15,14,16)(17,21,23,35)(18,26,27,42)(19,29,22,36)(20,31,33,34)(24,38,37,48)(25,39,40,41)(28,43,32,44)(30,45,46,47);(1,4,5,2)(3,7,8,6)(9,12,13,11)(10,15,14,16)(21,35)(26,42)(29,36)(31,34)(38,48)(39,41)(43,44)(45,47)Polycyclic presentation:
2^9|a^2=fhi,b^2=i,c^2=gi,d^2=fi,e^2=f,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=dhi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6957813);