Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7234437 |
| 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 | #2407 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1473 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(3,7)(17,18,22,28)(19,30,34,24)(20,26,36,44)(21,29,23,27)(25,38,42,35)(31,40,32,43)(33,41,37,45)(39,48,46,47);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16)(17,19)(18,24)(20,31)(21,37)(22,34)(23,33)(25,39)(26,43)(27,45)(28,30)(29,41)(32,36)(35,48)(38,47)(40,44)(42,46);(9,11)(10,13)(12,16)(14,15)(17,20,22,36)(18,25,28,42)(19,31,34,32)(21,35,23,38)(24,39,30,46)(26,29,44,27)(33,47,37,48)(40,45,43,41);(2,5)(6,8)(11,16)(13,15)(17,20)(18,26)(19,32)(21,38)(22,36)(23,35)(24,40)(25,27)(28,44)(29,42)(30,43)(31,34)(33,47)(37,48)(39,41)(45,46);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,21,22,23)(18,27,28,29)(19,33,34,37)(20,35,36,38)(24,41,30,45)(25,26,42,44)(31,47,32,48)(39,40,46,43)Polycyclic presentation:
2^9|a^2=i,c^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,g^c=gi,e^d=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7146053);