Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996375 |
| 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 | #23857 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1645 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(17,18,23,38,22,28,21,27)(19,29,37,24,31,25,32,41)(20,26,39,48,36,45,35,44)(30,43,47,40,34,42,33,46);(9,10)(11,15)(12,14)(13,16)(17,19,22,31)(18,24,28,41)(20,33,36,47)(21,37,23,32)(25,38,29,27)(26,43,45,42)(30,39,34,35)(40,48,46,44);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,19)(18,25)(20,34)(21,32)(22,31)(23,37)(24,27)(26,43)(28,29)(30,36)(33,39)(35,47)(38,41)(40,48)(42,45)(44,46);(11,16)(13,15)(17,20,23,39,22,36,21,35)(18,26,38,48,28,45,27,44)(19,30,37,47,31,34,32,33)(24,40,25,42,41,46,29,43);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(19,31)(24,41)(25,29)(30,34)(32,37)(33,47)(40,46)(42,43)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=gh,d^2=fi,e^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6998070);