Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #222256 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1228 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,6)(4,8)(5,7)(9,10)(11,19,23,15)(12,24,26,16)(13,18)(14,17)(20,30,22,28)(21,29)(25,37,38,31)(27,32)(33,36)(34,39)(35,40);(1,3,5,4)(2,6,7,8)(9,11)(10,15)(12,21)(13,20)(14,23)(16,29)(17,19)(18,28)(22,27)(24,33)(25,34)(26,36)(30,32)(31,39)(35,38)(37,40);(1,4,5,3)(2,6,7,8)(11,20)(13,27)(15,28)(18,32)(19,30)(21,34)(22,23)(25,38)(29,39)(31,37)(33,40)(35,36);(9,12)(10,16)(11,21)(13,25)(14,26)(15,29)(17,24)(18,31)(19,33)(20,34)(22,35)(23,36)(27,38)(28,39)(30,40)(32,37)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=f,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi,h^b=hi,g^c=gi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 268051);