Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806623 |
| 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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1635 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)(7,14)(8,17)(9,12)(10,13)(11,16)(15,19)(18,20)(21,22,27,33)(23,34,37,29)(24,30,35,43)(25,39,28,31)(26,32,41,48)(36,49,51,44)(38,45,50,47)(40,52,42,46);(5,7)(6,11)(8,15)(9,18)(10,16)(12,19)(13,14)(17,20)(21,23,27,37)(22,29,33,34)(24,25,35,28)(26,38,41,50)(30,31,43,39)(32,47,48,45)(36,42,51,40)(44,46,49,52);(1,2,3,4)(21,24,28,23,27,35,25,37)(22,30,31,34,33,43,39,29)(26,38,42,36,41,50,40,51)(32,45,46,49,48,47,52,44);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(21,25,27,28)(22,31,33,39)(23,35,37,24)(26,40,41,42)(29,43,34,30)(32,46,48,52)(36,50,51,38)(44,47,49,45);(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(21,26,28,42,27,41,25,40)(22,32,31,46,33,48,39,52)(23,36,35,50,37,51,24,38)(29,44,30,45,34,49,43,47)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=gh,d^2=f,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6723739);