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 | #23854 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1613 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)(17,18,24,28,23,30,22,29)(19,31,33,25,20,26,34,41)(21,27,37,46,39,45,38,44)(32,43,47,40,35,42,36,48);(9,10)(11,15)(12,14)(13,16)(17,19,23,20)(18,25,30,41)(21,36,39,47)(22,33,24,34)(26,28,31,29)(27,43,45,42)(32,37,35,38)(40,46,48,44);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,20,23,19)(18,26,30,31)(21,35,39,32)(22,33,24,34)(25,28,41,29)(27,42,45,43)(36,38,47,37)(40,46,48,44);(11,16)(13,15)(17,21,24,37,23,39,22,38)(18,27,28,46,30,45,29,44)(19,32,33,47,20,35,34,36)(25,40,26,42,41,48,31,43);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(17,22,23,24)(18,28,30,29)(19,33,20,34)(21,37,39,38)(25,31,41,26)(27,44,45,46)(32,36,35,47)(40,42,48,43)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=ghi,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=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6998106);