Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667932 |
| 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 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1695 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)(7,13)(8,12)(9,11)(10,15)(14,20)(16,18)(17,19)(21,22,27,31)(23,32,36,46)(24,29,37,43)(25,30,26,40)(28,34,42,35)(33,47,48,52)(38,44,39,45)(41,50,51,49);(1,2)(3,4)(5,7,9,15)(6,10,11,13)(8,16,17,14)(12,20,19,18)(24,37)(29,43)(33,48)(38,39)(41,51)(44,45)(47,52)(49,50);(2,4)(21,23,27,36)(22,28,31,42)(24,33,37,48)(25,35,26,34)(29,41,43,51)(30,46,40,32)(38,50,39,49)(44,47,45,52);(6,11)(10,13)(12,19)(18,20)(21,24)(22,29)(23,33)(25,39)(26,38)(27,37)(28,41)(30,45)(31,43)(32,47)(34,50)(35,49)(36,48)(40,44)(42,51)(46,52);(2,4)(5,8,9,17)(6,12,11,19)(7,14,15,16)(10,18,13,20)(21,25,27,26)(22,30,31,40)(23,34,36,35)(24,38,37,39)(28,32,42,46)(29,44,43,45)(33,49,48,50)(41,52,51,47)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^c=ei,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6581125);