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 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1657 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(5,6,9,12)(7,13,15,10)(8,16,17,11)(14,18,20,19)(21,22)(23,28)(24,34)(25,30)(26,40)(27,31)(29,36)(32,43)(33,41)(35,46)(37,47)(38,44)(39,50)(42,49)(45,51)(48,52);(2,3)(5,7)(6,10)(8,14)(9,15)(11,18)(12,13)(16,19)(17,20)(21,23,27,26)(22,28,31,40)(24,32,37,36)(25,33,38,39)(29,34,43,47)(30,41,44,50)(35,45,48,49)(42,46,51,52);(1,2,4,3)(21,24,23,32,27,37,26,36)(22,29,40,47,31,43,28,34)(25,35,33,45,38,48,39,49)(30,42,50,52,44,51,41,46);(5,8,9,17)(6,11,12,16)(7,14,15,20)(10,18,13,19)(21,25)(22,30)(23,33)(24,35)(26,39)(27,38)(28,41)(29,42)(31,44)(32,45)(34,46)(36,49)(37,48)(40,50)(43,51)(47,52);(1,3,4,2)(7,15)(10,13)(14,20)(18,19)(25,38)(30,44)(33,39)(35,48)(41,50)(42,51)(45,49)(46,52)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=ghi,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,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, 6722591);