Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469989 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1656 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)(7,10)(8,11)(9,12)(13,17)(14,18)(15,19)(16,20)(21,22,27,30)(23,32,35,43)(24,29,36,46)(25,31,26,39)(28,40,42,34)(33,47,48,52)(37,44,38,45)(41,49,51,50);(1,2,4,3)(5,7)(6,10)(8,15)(9,13)(11,19)(12,17)(14,16)(18,20)(21,23)(22,28)(24,33)(25,34)(26,40)(27,35)(29,41)(30,42)(31,43)(32,39)(36,48)(37,49)(38,50)(44,47)(45,52)(46,51);(1,3,4,2)(7,13)(10,17)(14,15)(18,19)(21,24)(22,29)(23,33)(25,38)(26,37)(27,36)(28,41)(30,46)(31,45)(32,47)(34,50)(35,48)(39,44)(40,49)(42,51)(43,52);(5,8,9,16)(6,11,12,20)(7,14,13,15)(10,18,17,19)(22,30)(24,36)(28,42)(31,39)(32,43)(33,48)(37,38)(49,50);(6,12)(10,17)(11,20)(18,19)(21,25,27,26)(22,31,30,39)(23,34,35,40)(24,37,36,38)(28,43,42,32)(29,44,46,45)(33,49,48,50)(41,47,51,52)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=g,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6391973);