Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806647 |
| 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×#547 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1774 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)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,22,27,32)(23,34,36,28)(24,38,37,29)(25,33,39,30)(26,31,43,48)(35,50,51,44)(40,47,42,45)(41,49,52,46);(5,7)(6,10)(8,14)(9,15)(11,17)(12,13)(16,20)(18,19)(21,23,27,36)(22,28,32,34)(24,39,37,25)(26,40,43,42)(29,30,38,33)(31,45,48,47)(35,41,51,52)(44,49,50,46);(2,4)(5,8,9,16)(6,11,12,18)(7,14,15,20)(10,17,13,19)(21,24,25,23,27,37,39,36)(22,29,33,28,32,38,30,34)(26,42,41,51,43,40,52,35)(31,47,49,50,48,45,46,44);(6,12)(10,13)(11,18)(17,19)(21,25,27,39)(22,30,32,33)(23,24,36,37)(26,41,43,52)(28,38,34,29)(31,46,48,49)(35,40,51,42)(44,47,50,45);(7,15)(10,13)(14,20)(17,19)(21,26,25,41,27,43,39,52)(22,31,33,49,32,48,30,46)(23,35,37,42,36,51,24,40)(28,44,38,47,34,50,29,45)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fhi,d^2=i,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=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6714513);