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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1760 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)(21,22,28,42,27,33,26,32)(23,34,36,29,24,30,37,45)(25,31,43,52,41,49,40,48)(35,47,50,44,38,46,39,51);(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23,27,24)(22,29,33,45)(25,39,41,50)(26,36,28,37)(30,42,34,32)(31,47,49,46)(35,43,38,40)(44,52,51,48);(1,2,3,4)(5,8)(6,11)(7,13)(9,16)(10,17)(12,20)(14,15)(18,19)(21,24,27,23)(22,30,33,34)(25,38,41,35)(26,36,28,37)(29,42,45,32)(31,46,49,47)(39,40,50,43)(44,52,51,48);(8,16)(11,20)(13,15)(17,19)(21,25,28,43,27,41,26,40)(22,31,42,52,33,49,32,48)(23,35,36,50,24,38,37,39)(29,44,30,46,45,51,34,47);(6,12)(7,14)(11,20)(13,15)(21,26,27,28)(22,32,33,42)(23,36,24,37)(25,40,41,43)(29,30,45,34)(31,48,49,52)(35,50,38,39)(44,46,51,47)Polycyclic presentation:
2^9|a^2=fhi,b^2=i,c^2=gi,d^2=fi,e^2=i,f^2=i,b^a=bf,c^a=cg,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, 6949398);