Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18446 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#386 |
| prime radical | #36652 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1302 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,9,11,16)(4,8)(5,7)(6,10,14,12)(13,15)(17,18)(19,22)(20,27,28,23)(21,24)(25,29,31,32)(26,30);(1,3,4,12,5,11,13,10)(2,6,8,16,7,14,15,9)(17,19)(18,22)(20,25)(21,26)(23,29)(24,30)(27,32)(28,31);(1,2)(3,6)(4,8)(5,7)(9,10)(11,14)(12,16)(13,15)(17,20)(18,23)(19,25)(21,28)(22,29)(24,27)(26,31)(30,32);(2,7)(6,14)(8,15)(9,16)(18,24)(22,30)(23,27)(29,32)Polycyclic presentation:
2^9|a^2=g,b^2=eg,e^2=h,b^a=be,c^a=cf,d^a=dhi,f^a=fi,e^b=eh,f^b=fi,g^b=gh,d^c=dh,e^c=ei,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 101893);