Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #162486 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #35081 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1035 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)(4,5)(7,8)(10,14)(11,18)(12,15)(16,23)(17,21)(20,24)(25,26)(27,30,34,38)(28,35)(29,32)(31,36)(33,39,40,37);(9,10,15,21,13,17,12,14)(11,16,22,23,18,24,19,20)(25,27,29,34)(26,30)(28,33,36,40)(31,37)(32,38)(35,39);(2,5)(6,8)(9,11)(10,16)(12,19)(13,18)(14,20)(15,22)(17,24)(21,23)(25,28)(26,31)(27,33)(29,36)(30,37)(32,35)(34,40)(38,39);(1,3,4,7)(2,6,5,8)(11,18)(16,24)(19,22)(20,23)(26,32)(28,36)(30,38)(33,40)Polycyclic presentation:
2^9|a^2=g,b^2=egh,d^2=fh,e^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,e^b=ei,g^b=gi,d^c=dh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 115125);