Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| 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 | #45412 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1776 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)(21,22,28,32,27,34,26,33)(23,30,36,29,24,38,37,46)(25,40,42,48,44,31,43,49)(35,50,51,52,39,47,41,45);(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(21,23,27,24)(22,29,34,46)(25,41,44,51)(26,36,28,37)(30,33,38,32)(31,47,40,50)(35,42,39,43)(45,49,52,48);(1,2,3,4)(5,8)(6,11)(7,13)(9,16)(10,17)(12,20)(14,15)(18,19)(21,24,27,23)(22,30,34,38)(25,39,44,35)(26,36,28,37)(29,33,46,32)(31,47,40,50)(41,43,51,42)(45,48,52,49);(8,16)(11,20)(13,15)(17,19)(21,25,28,42,27,44,26,43)(22,31,32,49,34,40,33,48)(23,35,36,51,24,39,37,41)(29,45,38,50,46,52,30,47);(6,12)(7,14)(11,20)(13,15)(21,26,27,28)(22,32,34,33)(23,36,24,37)(25,42,44,43)(29,30,46,38)(31,48,40,49)(35,41,39,51)(45,50,52,47)Polycyclic presentation:
2^9|a^2=fhi,b^2=i,c^2=g,d^2=fi,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6949810);