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 | #2535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1590 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)(17,18,23,29,24,30,22,28)(19,26,32,41,20,33,35,25)(21,27,38,44,39,45,37,43)(31,42,36,48,34,46,47,40);(9,10)(11,15)(12,14)(13,16)(17,19,24,20)(18,25,30,41)(21,36,39,47)(22,32,23,35)(26,29,33,28)(27,42,45,46)(31,37,34,38)(40,43,48,44);(1,3,5,4)(2,6,7,8)(9,11,12,16)(10,13,14,15)(17,20,24,19)(18,26,30,33)(21,34,39,31)(22,32,23,35)(25,29,41,28)(27,42,45,46)(36,38,47,37)(40,44,48,43);(11,16)(13,15)(17,21,22,37,24,39,23,38)(18,27,28,43,30,45,29,44)(19,31,35,47,20,34,32,36)(25,40,33,46,41,48,26,42);(1,4,5,3)(2,6,7,8)(10,14)(13,15)(17,22,24,23)(18,28,30,29)(19,32,20,35)(21,37,39,38)(25,26,41,33)(27,43,45,44)(31,36,34,47)(40,42,48,46)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=gh,d^2=fi,e^2=g,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6997925);