Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #396363 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14732 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23402 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1452 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)(3,12)(4,9,17,25)(5,19)(6,10)(7,11)(8,14)(13,26,28,32)(15,22)(16,29,18,24)(20,23)(21,30,31,27);(1,3,5,14,7,15,6,20)(2,8,10,12,11,23,19,22)(4,13,16,27,17,28,18,30)(9,21,24,26,25,31,29,32);(1,2)(3,12)(4,9)(5,10)(6,19)(7,11)(8,20)(13,26)(14,23)(15,22)(16,24)(17,25)(18,29)(21,30)(27,31)(28,32);(1,4,5,16,7,17,6,18)(2,9,10,24,11,25,19,29)(3,13,14,27,15,28,20,30)(8,21,12,26,23,31,22,32)Polycyclic presentation:
2^9|a^2=f,b^2=e,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,g^b=gi,g^c=gi,h^c=hi,f^d=fi,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72213603, #72215931, #72215974, … |
| maximal normal subgroup of | #72213603, #72215931, #72215974, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 303559);