Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #481819 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C22⋊C4 | C24 |
| prime radical | C2×#2631 | C2 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1252 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)(3,9)(4,17,19,7)(5,8)(6,12)(10,18,21,29)(11,14)(13,26)(15,23,27,20)(16,22)(24,31,32,30)(25,28);(2,7)(3,13)(4,18)(8,10)(9,23)(11,21)(12,17)(15,30)(16,28)(19,29)(20,26)(22,32)(24,25)(27,31);(2,8)(7,10)(9,22)(11,12)(17,21)(20,24)(23,32)(25,26);(1,3,14,28)(2,9,11,25)(4,15,29,30)(5,16,6,13)(7,20,21,32)(8,22,12,26)(10,24,17,23)(18,31,19,27)Polycyclic presentation:
2^9|a^2=f,d^2=hi,e^2=h,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 | #72231335, #72231342, #72231369, … |
| maximal normal subgroup of | #72231335, #72231342, #72231369, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 389158);