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 | C2×#3179 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1736 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)(21,22,26,30)(23,27,33,39)(24,28,36,42)(25,29,32,38)(31,37,43,47)(34,40,46,50)(35,41,45,49)(44,48,51,52);(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20)(23,31)(24,35)(25,32)(27,37)(28,41)(29,38)(33,43)(34,46)(36,45)(39,47)(40,50)(42,49);(2,4)(21,23,32,43,26,33,25,31)(22,27,38,47,30,39,29,37)(24,34,45,51,36,46,35,44)(28,40,49,52,42,50,41,48);(5,8,9,17)(6,11,12,16)(7,14,15,20)(10,18,13,19)(23,33)(27,39)(31,43)(34,46)(37,47)(40,50)(44,51)(48,52);(7,15)(10,13)(14,20)(18,19)(21,24,25,35,26,36,32,45)(22,28,29,41,30,42,38,49)(23,34,31,44,33,46,43,51)(27,40,37,48,39,50,47,52)Polycyclic presentation:
2^9|a^2=fi,c^2=hi,d^2=f,e^2=h,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, 6715773);