Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #45136 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1780 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,4)(5,6,9,12)(7,10,14,18)(8,11,16,20)(13,17,15,19)(21,22,28,43,27,32,26,31)(23,33,25,29,35,49,36,46)(24,37,44,48,40,30,41,52)(34,51,42,47,38,50,39,45);(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23,27,35)(22,29,32,46)(24,38,40,34)(25,28,36,26)(30,47,37,45)(31,49,43,33)(39,44,42,41)(48,50,52,51);(1,2,3,4)(5,8)(6,11)(7,13)(9,16)(10,17)(12,20)(14,15)(18,19)(21,24,27,40)(22,30,32,37)(23,34,35,38)(25,42,36,39)(26,41,28,44)(29,45,46,47)(31,48,43,52)(33,50,49,51);(8,16)(11,20)(13,15)(17,19)(21,25,26,23,27,36,28,35)(22,29,31,33,32,46,43,49)(24,39,41,38,40,42,44,34)(30,47,48,51,37,45,52,50);(6,12)(7,14)(11,20)(13,15)(23,35)(24,40)(25,36)(29,46)(30,37)(33,49)(41,44)(48,52)Polycyclic presentation:
2^9|a^2=fh,b^2=i,c^2=g,d^2=f,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, 6949535);