Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #482075 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#102 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | #60673 | C2 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
805 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)(3,9,11,6)(4,8)(5,7)(10,15,16,13)(12,14)(18,23,25,31)(19,21)(20,27)(24,32,26,28)(29,30);(1,3)(2,6)(4,10)(5,11)(7,9)(8,13)(12,16)(14,15)(17,18)(19,24)(20,28)(21,26)(22,25)(23,29)(27,31)(30,32);(2,7)(6,9)(8,14)(13,15)(17,19,22,21)(18,24,25,26)(20,27,30,29)(23,28,31,32);(1,4,5,12)(2,8,7,14)(3,10,11,16)(6,13,9,15)(17,19,22,21)(18,24,25,26)(20,27,30,29)(23,28,31,32)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=ghi,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cghi,d^a=di,e^a=eg,g^a=gi,h^a=hi,e^b=eh,f^b=fghi,g^b=gi,h^b=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72319589, #72319660, #72319677, … |
| maximal normal subgroup of | #72319589, #72319660, #72319677, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 389528);