Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682460 |
| 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 | #2407 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1732 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(5,6)(7,14)(8,12)(9,18)(10,13)(11,17)(15,20)(16,19)(21,22,26,30)(23,27,32,40)(24,34,35,49)(25,37,38,29)(28,42,43,36)(31,44,46,52)(33,48,47,41)(39,50,51,45);(2,3)(5,7)(6,11)(8,15)(9,16)(10,17)(12,19)(13,14)(18,20)(21,23,26,32)(22,27,30,40)(24,31,35,46)(25,33,38,47)(28,39,43,51)(29,41,37,48)(34,44,49,52)(36,45,42,50);(1,2,4,3)(5,8,10,9)(6,12,13,18)(7,15,17,16)(11,19,14,20)(21,24,25,36,26,35,38,42)(22,28,29,34,30,43,37,49)(23,31,33,45,32,46,47,50)(27,39,41,44,40,51,48,52);(5,9,10,8)(6,12,13,18)(7,16,17,15)(11,19,14,20)(23,32)(24,35)(27,40)(28,43)(33,47)(34,49)(36,42)(41,48);(1,3,4,2)(7,17)(11,14)(15,16)(19,20)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fgh,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6722900);