Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535655 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#546 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2061 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,4)(5,6,9,12)(7,10,13,16)(8,11,15,18)(14,17,19,20)(21,22,27,31)(23,32,36,47)(24,29,39,45)(25,30,26,40)(28,34,42,35)(33,46,50,51)(37,43,38,44)(41,49,52,48);(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23,25,34,27,36,26,35)(22,28,40,47,31,42,30,32)(24,33,38,49,39,50,37,48)(29,41,43,51,45,52,44,46);(1,2,3,4)(7,13)(10,16)(14,19)(17,20)(21,24,27,39)(22,29,31,45)(23,33,36,50)(25,38,26,37)(28,41,42,52)(30,44,40,43)(32,46,47,51)(34,49,35,48);(5,8)(6,11)(7,14)(9,15)(10,17)(12,18)(13,19)(16,20)(21,25,27,26)(22,30,31,40)(23,34,36,35)(24,37,39,38)(28,32,42,47)(29,43,45,44)(33,48,50,49)(41,51,52,46);(8,15)(11,18)(14,19)(17,20)(21,25,27,26)(22,30,31,40)(23,34,36,35)(24,38,39,37)(28,32,42,47)(29,44,45,43)(33,49,50,48)(41,46,52,51)Polycyclic presentation:
2^9|a^2=hi,b^2=fi,c^2=gi,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^c=di,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6476952);