Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682360 |
| 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 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1661 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(5,6)(7,14)(8,17)(9,12)(10,13)(11,16)(15,19)(18,20)(21,22,25,27)(23,31,32,45)(24,36,37,29)(26,39,33,30)(28,41,40,35)(34,48,46,44)(38,43,47,51)(42,49,50,52);(1,2)(3,4)(5,7)(6,11)(8,15)(9,18)(10,16)(12,19)(13,14)(17,20)(22,27)(23,32)(24,37)(30,39)(35,41)(38,47)(42,50)(44,48);(2,4)(21,23,33,41,25,32,26,35)(22,28,39,45,27,40,30,31)(24,34,47,52,37,46,38,49)(29,42,51,48,36,50,43,44);(5,8,10,9)(6,12,13,17)(7,15,16,18)(11,19,14,20)(21,24,25,37)(22,29,27,36)(23,34,32,46)(26,38,33,47)(28,42,40,50)(30,43,39,51)(31,44,45,48)(35,49,41,52);(2,4)(5,9,10,8)(6,12,13,17)(7,15,16,18)(11,20,14,19)(24,37)(29,36)(34,46)(38,47)(42,50)(43,51)(44,48)(49,52)Polycyclic presentation:
2^9|a^2=i,c^2=gi,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6593727);