Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996355 |
| 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 | #45385 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1782 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,4)(5,6,9,12)(7,10,14,18)(8,11,16,20)(13,17,15,19)(21,22,27,32,28,33,26,44)(23,34,36,31,37,50,25,29)(24,38,41,48,42,30,45,49)(35,52,40,46,39,51,43,47);(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23,28,37)(22,29,33,31)(24,39,42,35)(25,26,36,27)(30,47,38,46)(32,50,44,34)(40,45,43,41)(48,52,49,51);(1,2,3,4)(5,8)(6,11)(7,13)(9,16)(10,17)(12,20)(14,15)(18,19)(21,24)(22,30)(23,35)(25,43)(26,45)(27,41)(28,42)(29,46)(31,47)(32,49)(33,38)(34,51)(36,40)(37,39)(44,48)(50,52);(8,16)(11,20)(13,15)(17,19)(21,25,27,23,28,36,26,37)(22,31,32,50,33,29,44,34)(24,40,41,39,42,43,45,35)(30,46,49,51,38,47,48,52);(6,12)(7,14)(11,20)(13,15)(21,26,28,27)(22,32,33,44)(23,25,37,36)(24,41,42,45)(29,34,31,50)(30,48,38,49)(35,40,39,43)(46,52,47,51)Polycyclic presentation:
2^9|a^2=fhi,b^2=i,c^2=gi,d^2=f,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6949652);