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)(21,22,27,32)(23,28,36,44)(24,30,38,45)(25,31,26,40)(29,39,46,37)(33,42,48,51)(34,43,35,50)(41,49,52,47);(2,3)(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23,27,36)(22,28,32,44)(24,33,38,48)(25,34,26,35)(29,41,46,52)(30,42,45,51)(31,43,40,50)(37,47,39,49);(5,8)(6,11)(7,13)(9,16)(10,17)(12,20)(14,15)(18,19)(21,24,27,38)(22,29,32,46)(23,33,36,48)(25,37,26,39)(28,41,44,52)(30,31,45,40)(34,47,35,49)(42,43,51,50);(1,3,4,2)(8,16)(11,20)(13,15)(17,19)(21,24)(22,30)(23,33)(25,39)(26,37)(27,38)(28,42)(29,40)(31,46)(32,45)(34,49)(35,47)(36,48)(41,50)(43,52)(44,51);(6,12)(7,14)(11,20)(13,15)(21,25,27,26)(22,31,32,40)(23,34,36,35)(24,37,38,39)(28,43,44,50)(29,45,46,30)(33,47,48,49)(41,51,52,42)Polycyclic presentation:
2^9|a^2=fhi,b^2=i,c^2=i,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=eh,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, 6941964);