Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7300980 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1814 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)(3,6)(4,5)(7,8)(17,18)(19,24)(20,30)(21,33)(22,36)(23,27)(25,28)(26,35)(29,42)(31,45)(32,38)(34,39)(37,41)(40,46)(43,48)(44,47);(2,5)(3,7)(9,10)(11,15)(12,14)(13,16)(17,19,23,22)(18,24,27,36)(20,28,32,37)(21,34,35,29)(25,38,41,30)(26,42,33,39)(31,46,47,43)(40,44,48,45);(9,11)(10,13)(12,16)(14,15)(17,20,23,32)(18,25,27,41)(19,28,22,37)(21,31,35,47)(24,38,36,30)(26,40,33,48)(29,43,34,46)(39,45,42,44);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,21,23,35)(18,26,27,33)(19,29,22,34)(20,31,32,47)(24,39,36,42)(25,40,41,48)(28,43,37,46)(30,44,38,45);(2,5)(6,8)(10,14)(13,15)(21,35)(26,33)(29,34)(31,47)(39,42)(40,48)(43,46)(44,45)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=eh,d^c=dh,g^c=gi,e^d=ei59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7210126);