Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806671 |
| 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 | |
1767 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)(26,42,43,31)(35,44,49,48)(40,45,50,51)(41,46,52,47);(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)(26,40,43,50)(29,30,38,33)(31,45,42,51)(35,41,49,52)(44,47,48,46);(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)(26,40,41,35,43,50,52,49)(31,45,47,44,42,51,46,48);(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)(31,46,42,47)(35,40,49,50)(44,51,48,45);(7,15)(10,13)(14,20)(17,19)(21,26,39,52,27,43,25,41)(22,31,30,46,32,42,33,47)(23,35,24,40,36,49,37,50)(28,44,29,45,34,48,38,51)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fhi,d^2=i,e^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6714538);