Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806623 |
| 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 | #42683 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ 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)(7,14)(8,12)(9,18)(10,13)(11,17)(15,20)(16,19)(21,22,27,32)(23,28,34,41)(24,29,36,43)(25,30,38,45)(26,31,39,46)(33,40,48,51)(35,42,49,52)(37,44,47,50);(5,7)(6,11)(8,15)(9,16)(10,17)(12,19)(13,14)(18,20)(21,23,27,34)(22,28,32,41)(24,35,36,49)(25,26,38,39)(29,42,43,52)(30,31,45,46)(33,47,48,37)(40,50,51,44);(1,2,3,4)(5,8,10,9)(6,12,13,18)(7,15,17,16)(11,19,14,20)(21,24,26,37,27,36,39,47)(22,29,31,44,32,43,46,50)(23,33,25,35,34,48,38,49)(28,40,30,42,41,51,45,52);(5,9,10,8)(6,12,13,18)(7,16,17,15)(11,19,14,20)(24,36)(29,43)(33,48)(35,49)(37,47)(40,51)(42,52)(44,50);(7,17)(11,14)(15,16)(19,20)(21,25,26,34,27,38,39,23)(22,30,31,41,32,45,46,28)(24,35,37,48,36,49,47,33)(29,42,44,51,43,52,50,40)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fgh,d^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6722703);