Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #481819 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C4⋊C4 | C24 |
| prime radical | C2×#2635 | C2 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1193 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,12,6,13,15,29)(3,10,16,25,14,30,18,8)(4,11,17,26,19,31,21,32)(7,9,22,27,23,28,20,24);(2,8)(3,14)(4,18)(5,20)(9,11)(10,27)(12,26)(13,25)(15,22)(16,19)(17,21)(24,30)(28,31)(29,32);(2,9)(8,11)(10,26)(12,27)(13,28)(24,29)(25,31)(30,32);(1,3,7,17)(2,10,9,26)(4,15,18,20)(5,16,22,19)(6,14,23,21)(8,24,11,29)(12,25,27,31)(13,30,28,32)Polycyclic presentation:
2^9|a^2=f,d^2=h,e^2=h,f^2=i,b^a=be,c^a=ch,e^a=eg,g^a=gi,d^b=di,e^b=eh,f^b=fgh,f^e=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72233278, #72233309, #72233311, … |
| maximal normal subgroup of | #72233278, #72233309, #72233311, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 389181);