Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806647 |
| 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 | |
1769 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)(21,22,26,31)(23,28,34,43)(24,36,39,29)(25,30,35,44)(27,32,38,46)(33,49,51,42)(37,48,41,45)(40,52,50,47);(5,7)(6,11)(8,15)(9,16)(10,17)(12,19)(13,14)(18,20)(21,23,26,34)(22,28,31,43)(24,37,39,41)(25,38,35,27)(29,45,36,48)(30,46,44,32)(33,50,51,40)(42,52,49,47);(1,2,3,4)(5,8,10,9)(6,12,13,18)(7,15,17,16)(11,19,14,20)(21,24,38,50,26,39,27,40)(22,29,46,52,31,36,32,47)(23,33,25,41,34,51,35,37)(28,42,30,48,43,49,44,45);(5,9,10,8)(6,12,13,18)(7,16,17,15)(11,19,14,20)(24,39)(29,36)(33,51)(37,41)(40,50)(42,49)(45,48)(47,52);(7,17)(11,14)(15,16)(19,20)(21,25,27,23,26,35,38,34)(22,30,32,28,31,44,46,43)(24,37,40,51,39,41,50,33)(29,45,47,49,36,48,52,42)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=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6722735);