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 | |
1773 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)(23,32)(24,35)(25,30)(26,31)(27,28)(29,33)(34,45)(36,44)(37,48)(38,42)(39,41)(40,46)(43,47)(49,52)(50,51);(2,3)(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23,26,27)(22,28,31,32)(24,36,38,33)(25,39,40,34)(29,42,44,35)(30,45,46,41)(37,47,50,49)(43,48,52,51);(5,8)(6,11)(7,13)(9,16)(10,17)(12,20)(14,15)(18,19)(21,24,26,38)(22,29,31,44)(23,33,27,36)(25,37,40,50)(28,35,32,42)(30,43,46,52)(34,47,39,49)(41,48,45,51);(1,3,4,2)(8,16)(11,20)(13,15)(17,19)(21,25,26,40)(22,30,31,46)(23,34,27,39)(24,37,38,50)(28,41,32,45)(29,43,44,52)(33,47,36,49)(35,48,42,51);(6,12)(7,14)(11,20)(13,15)(22,31)(25,40)(28,32)(29,44)(34,39)(35,42)(37,50)(47,49)Polycyclic presentation:
2^9|a^2=fh,b^2=i,c^2=i,d^2=f,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, 6942011);