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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1692 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)(23,34)(24,38)(25,31)(26,32)(27,33)(28,43)(29,37)(30,39)(35,49)(36,47)(40,45)(41,48)(42,46)(44,51)(50,52);(2,3)(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23,27,37)(22,29,33,34)(24,35,25,40)(26,42,28,36)(30,44,48,50)(31,49,38,45)(32,47,43,46)(39,52,41,51);(5,8,9,16)(6,11,12,20)(7,13,14,15)(10,17,18,19)(21,24)(22,30)(23,35)(25,27)(26,39)(28,41)(29,44)(31,32)(33,48)(34,50)(36,51)(37,40)(38,43)(42,52)(45,46)(47,49);(1,3,4,2)(8,16)(11,20)(13,15)(17,19)(21,25,27,24)(22,31,33,38)(23,35,37,40)(26,39,28,41)(29,45,34,49)(30,43,48,32)(36,52,42,51)(44,47,50,46);(6,12)(7,14)(11,20)(13,15)(21,26,27,28)(22,32,33,43)(23,36,37,42)(24,39,25,41)(29,46,34,47)(30,31,48,38)(35,51,40,52)(44,45,50,49)Polycyclic presentation:
2^9|a^2=fhi,b^2=i,c^2=h,d^2=f,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, 7011800);