Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535664 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1638 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2)(3,4)(21,22,27,31)(23,32,36,43)(24,29,40,48)(25,30,26,41)(28,35,44,34)(33,49,37,52)(38,46,39,47)(42,50,45,51);(5,6)(7,11)(8,12)(9,10)(13,19)(14,17)(15,20)(16,18)(21,23,25,35,27,36,26,34)(22,28,41,43,31,44,30,32)(24,37,39,51,40,33,38,50)(29,45,46,49,48,42,47,52);(2,4)(6,10)(11,17)(12,18)(19,20)(21,24)(22,29)(23,33)(25,39)(26,38)(27,40)(28,42)(30,47)(31,48)(32,49)(34,51)(35,50)(36,37)(41,46)(43,52)(44,45);(5,7,9,14)(6,11,10,17)(8,15,16,13)(12,20,18,19)(21,25,27,26)(22,30,31,41)(23,34,36,35)(24,38,40,39)(28,43,44,32)(29,46,48,47)(33,50,37,51)(42,49,45,52);(5,8,9,16)(6,12,10,18)(7,13,14,15)(11,19,17,20)(21,25,27,26)(22,30,31,41)(23,35,36,34)(24,39,40,38)(28,32,44,43)(29,47,48,46)(33,50,37,51)(42,49,45,52)Polycyclic presentation:
2^9|a^2=i,b^2=fi,d^2=h,e^2=h,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,d^c=di,e^d=ehi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6444245);