Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #222240 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1118 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)(9,10,13,17,15,20,14,19)(11,21,25,32,22,31,24,16)(12,26,28,33,29,34,27,18)(23,30,37,35,36,40,38,39);(2,5)(6,7)(9,11,13,25,15,22,14,24)(10,16,17,21,20,32,19,31)(12,23,28,37,29,36,27,38)(18,30,26,35,33,40,34,39);(10,17)(11,22)(13,14)(16,21)(18,26)(19,20)(23,36)(27,28)(30,35)(31,32)(33,34)(39,40);(1,3,4,8)(2,6,5,7)(9,12)(10,18)(11,23)(13,28)(14,27)(15,29)(16,30)(17,26)(19,34)(20,33)(21,35)(22,36)(24,38)(25,37)(31,39)(32,40)Polycyclic presentation:
2^9|a^2=h,b^2=h,d^2=efg,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,e^b=ei,g^b=gi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 279822);