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)(21,22,28,41,27,32,26,31)(23,33,25,29,36,43,35,44)(24,30,38,45,40,47,39,46)(34,48,37,42,50,52,49,51);(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23,27,36)(22,29,32,44)(24,34,40,50)(25,28,35,26)(30,42,47,51)(31,43,41,33)(37,38,49,39)(45,48,46,52);(1,2,3,4)(5,8)(6,11)(7,13)(9,16)(10,17)(12,20)(14,15)(18,19)(21,24,27,40)(22,30,32,47)(23,34,36,50)(25,37,35,49)(26,39,28,38)(29,42,44,51)(31,46,41,45)(33,48,43,52);(8,16)(11,20)(13,15)(17,19)(21,25,26,23,27,35,28,36)(22,29,31,33,32,44,41,43)(24,37,39,34,40,49,38,50)(30,42,46,48,47,51,45,52);(6,12)(7,14)(11,20)(13,15)(21,26,27,28)(22,31,32,41)(23,25,36,35)(24,38,40,39)(29,43,44,33)(30,45,47,46)(34,49,50,37)(42,48,51,52)Polycyclic presentation:
2^9|a^2=fhi,b^2=i,c^2=gi,d^2=f,e^2=i,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=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6949428);