Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| 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 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1649 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2,4,3)(8,16)(11,17)(15,20)(18,19)(21,22,28,32,27,34,26,33)(23,29,42,45,38,47,37,46)(24,39,40,30,25,31,41,48)(35,49,50,43,36,44,51,52);(2,3)(5,6,9,12)(7,13,14,10)(8,17,16,11)(15,18,20,19)(21,23,27,38)(22,29,34,47)(24,35,25,36)(26,37,28,42)(30,43,48,52)(31,44,39,49)(32,45,33,46)(40,50,41,51);(5,7)(6,10)(8,15)(9,14)(11,18)(12,13)(16,20)(17,19)(21,24,27,25)(22,30,34,48)(23,35,38,36)(26,40,28,41)(29,43,47,52)(31,32,39,33)(37,50,42,51)(44,45,49,46);(1,3,4,2)(7,14)(10,13)(15,20)(18,19)(21,25,27,24)(22,31,34,39)(23,36,38,35)(26,40,28,41)(29,44,47,49)(30,32,48,33)(37,50,42,51)(43,45,52,46);(5,8,9,16)(6,11,12,17)(7,15,14,20)(10,18,13,19)(21,26,27,28)(22,32,34,33)(23,37,38,42)(24,40,25,41)(29,45,47,46)(30,39,48,31)(35,50,36,51)(43,49,52,44)Polycyclic presentation:
2^9|a^2=fgi,b^2=hi,c^2=i,d^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ehi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7091036);