Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996355 |
| 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×#127 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1712 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)(23,31)(24,35)(25,30)(26,32)(27,28)(29,38)(33,45)(34,46)(36,51)(37,42)(39,43)(40,52)(41,48)(44,50)(47,49);(2,3)(5,6,9,12)(7,13,14,10)(8,17,16,11)(15,18,20,19)(21,23,26,27)(22,28,32,31)(24,33,37,38)(25,39,41,34)(29,42,45,35)(30,46,48,43)(36,50,40,49)(44,51,47,52);(5,7)(6,10)(8,15)(9,14)(11,18)(12,13)(16,20)(17,19)(21,24,27,38,26,37,23,33)(22,29,31,35,32,45,28,42)(25,40,39,49,41,36,34,50)(30,47,46,52,48,44,43,51);(1,3,4,2)(7,14)(10,13)(15,20)(18,19)(21,25)(22,30)(23,34)(24,36)(26,41)(27,39)(28,43)(29,44)(31,46)(32,48)(33,49)(35,51)(37,40)(38,50)(42,52)(45,47);(5,8,9,16)(6,11,12,17)(7,15,14,20)(10,18,13,19)(21,23,26,27)(22,31,32,28)(24,33,37,38)(25,39,41,34)(29,35,45,42)(30,43,48,46)(36,50,40,49)(44,52,47,51)Polycyclic presentation:
2^9|a^2=fi,b^2=hi,c^2=g,d^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7093764);