Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20286 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | #2358 |
| prime radical | C2×#41 | C2×D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
628 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,7)(3,10,12,16)(4,9,6,14)(8,11,15,13)(17,18,23,27,24,29,22,28)(19,20,30,25,21,26,32,31);(2,7)(8,15)(9,14)(10,16)(17,19,22,32,24,21,23,30)(18,25,28,20,29,31,27,26);(1,3,5,12)(2,8,7,15)(4,13,6,11)(9,16,14,10)(17,20,24,26)(18,19,29,21)(22,31,23,25)(27,30,28,32);(1,4,5,6)(2,9,7,14)(3,11,12,13)(8,10,15,16)(17,21,24,19)(18,26,29,20)(22,30,23,32)(25,27,31,28)Polycyclic presentation:
2^9|a^2=g,b^2=gi,c^2=i,d^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dei,e^b=eh,d^c=dhi,f^c=fi,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 135433);