Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #188049 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C26 | C2×C4 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2762 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,10,5,9,17,23)(3,11,13,24,14,7,15,20)(6,8,16,22,18,21,29,31)(12,25,26,32,27,19,28,30);(1,3,5,14)(2,7,9,11)(4,15,17,13)(6,12,18,27)(8,19,21,25)(10,24,23,20)(16,28,29,26)(22,32,31,30);(2,8)(4,16)(7,19)(9,21)(11,25)(13,26)(15,28)(17,29);(3,12)(7,19)(11,25)(13,26)(14,27)(15,28)(20,30)(24,32)Polycyclic presentation:
2^9|a^2=g,b^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=di,g^b=gh,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72268070, #72268071, #72268072, … |
| maximal normal subgroup of | #72268070, #72268071, #72268072, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 95319);