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 | #42806 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1791 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)(23,28)(24,34)(25,30)(26,40)(27,31)(29,32)(33,42)(35,44)(36,47)(37,41)(38,45)(39,50)(43,46)(48,52)(49,51);(2,3)(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23,27,26)(22,28,31,40)(24,35,37,32)(25,33,38,39)(29,34,44,41)(30,42,45,50)(36,48,49,46)(43,47,52,51);(5,8)(6,11)(7,13)(9,16)(10,17)(12,20)(14,15)(18,19)(21,24,27,37)(22,29,31,44)(23,32,26,35)(25,36,38,49)(28,41,40,34)(30,43,45,52)(33,46,39,48)(42,51,50,47);(1,3,4,2)(8,16)(11,20)(13,15)(17,19)(21,25,27,38)(22,30,31,45)(23,33,26,39)(24,36,37,49)(28,42,40,50)(29,43,44,52)(32,46,35,48)(34,47,41,51);(6,12)(7,14)(11,20)(13,15)(22,31)(25,38)(28,40)(29,44)(33,39)(34,41)(36,49)(46,48)Polycyclic presentation:
2^9|a^2=fhi,b^2=i,c^2=i,d^2=fi,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, 6942029);