Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417738 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1084 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,5,4)(3,7,8,6)(9,10)(11,21)(12,18)(13,16)(14,19)(15,20)(17,24)(22,36)(23,32)(25,29)(26,34)(27,30)(28,33)(31,38)(35,39)(37,40);(1,3)(2,6)(4,7)(5,8)(10,16)(12,26)(13,14)(17,29)(18,33)(19,20)(21,32)(22,37)(23,24)(27,28)(35,36)(38,39);(1,4,5,2)(3,7,8,6)(9,11)(10,17)(12,22)(13,24)(14,23)(15,25)(16,29)(18,31)(19,21)(20,32)(26,38)(27,37)(28,39)(30,36)(33,40)(34,35);(3,8)(6,7)(9,12)(10,18)(11,22)(13,27)(14,26)(15,28)(16,30)(17,31)(19,34)(20,33)(21,35)(23,38)(24,37)(25,39)(29,36)(32,40)Polycyclic presentation:
2^9|a^2=ef,c^2=ef,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,f^b=fi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 332292);