Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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 | #42926 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1796 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)(5,6,9,12)(7,10,14,18)(8,11,16,20)(13,17,15,19)(21,22,26,32)(23,33,37,28)(24,30,39,47)(25,31,27,42)(29,41,48,38)(34,44,35,49)(36,46,40,45)(43,52,50,51);(2,3)(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23)(22,28)(24,34)(25,40)(26,37)(27,36)(29,43)(30,49)(31,46)(32,33)(35,39)(38,52)(41,51)(42,45)(44,47)(48,50);(5,8)(6,11)(7,13)(9,16)(10,17)(12,20)(14,15)(18,19)(21,24,26,39)(22,29,32,48)(23,34,37,35)(25,38,27,41)(28,43,33,50)(30,31,47,42)(36,51,40,52)(44,45,49,46);(1,3,4,2)(8,16)(11,20)(13,15)(17,19)(21,24)(22,30)(23,35)(25,41)(26,39)(27,38)(28,44)(29,42)(31,48)(32,47)(33,49)(34,37)(36,51)(40,52)(43,46)(45,50);(6,12)(7,14)(11,20)(13,15)(21,25,26,27)(22,31,32,42)(23,36,37,40)(24,38,39,41)(28,45,33,46)(29,47,48,30)(34,51,35,52)(43,49,50,44)Polycyclic presentation:
2^9|a^2=fh,c^2=i,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6942006);