Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| 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 | #1999 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1700 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2)(3,4)(5,6,9,12)(7,13,15,10)(8,16,17,11)(14,18,20,19)(21,22)(23,33)(24,37)(25,31)(26,30)(27,32)(28,35)(29,36)(34,49)(38,44)(39,47)(40,45)(41,46)(42,48)(43,50)(51,52);(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20)(21,23,26,35)(22,28,30,33)(24,38,36,27)(25,39,41,40)(29,44,37,32)(31,45,46,47)(34,42,50,51)(43,48,49,52);(2,4)(21,24,27,23,26,36,38,35)(22,29,32,28,30,37,44,33)(25,40,42,50,41,39,51,34)(31,47,48,49,46,45,52,43);(5,8,9,17)(6,11,12,16)(7,14,15,20)(10,18,13,19)(22,30)(25,41)(28,33)(29,37)(32,44)(34,50)(39,40)(42,51);(7,15)(10,13)(14,20)(18,19)(21,25,38,51,26,41,27,42)(22,31,44,52,30,46,32,48)(23,34,24,39,35,50,36,40)(28,43,29,45,33,49,37,47)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=h,d^2=fi,e^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,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, 6716038);