Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | D4×C22⋊C4 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1263 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)(3,7)(4,5)(6,8)(9,10,13,19,15,20,14,18)(11,21,24,33,25,32,23,16)(12,26,29,17,30,35,28,36)(22,38,39,34,40,31,27,37);(2,5)(6,7)(9,11)(10,16)(12,27)(13,24)(14,23)(15,25)(17,34)(18,32)(19,21)(20,33)(22,29)(26,38)(28,40)(30,39)(31,35)(36,37);(9,12)(10,17)(11,22)(13,29)(14,28)(15,30)(16,31)(18,26)(19,35)(20,36)(21,37)(23,27)(24,39)(25,40)(32,34)(33,38);(1,3,4,8)(2,6,5,7)(9,10)(11,16)(12,17)(13,19)(14,18)(15,20)(21,24)(22,31)(23,32)(25,33)(26,28)(27,34)(29,35)(30,36)(37,39)(38,40)Polycyclic presentation:
2^9|a^2=h,d^2=ef,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi,e^d=ei,f^d=fi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 268361);