Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533195 |
| 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×#553 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1705 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,3)(5,6,9,12)(7,10,13,16)(8,11,15,18)(14,17,19,20)(21,22,27,31)(23,28,34,43)(24,35,37,45)(25,30,26,39)(29,40,46,36)(32,47,49,52)(33,42,38,44)(41,48,51,50);(1,2,4,3)(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23,27,34)(22,28,31,43)(24,32,37,49)(25,38,26,33)(29,41,46,51)(30,44,39,42)(35,47,45,52)(36,50,40,48);(1,3,4,2)(7,13)(10,16)(14,19)(17,20)(21,24,27,37)(22,29,31,46)(23,32,34,49)(25,36,26,40)(28,41,43,51)(30,45,39,35)(33,48,38,50)(42,47,44,52);(5,8)(6,11)(7,14)(9,15)(10,17)(12,18)(13,19)(16,20)(21,25,27,26)(22,30,31,39)(23,33,34,38)(24,36,37,40)(28,42,43,44)(29,45,46,35)(32,48,49,50)(41,47,51,52);(8,15)(11,18)(14,19)(17,20)(23,34)(28,43)(32,49)(33,38)(41,51)(42,44)(47,52)(48,50)Polycyclic presentation:
2^9|a^2=hi,b^2=fi,c^2=fi,d^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ei,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6473450);