Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417980 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2670 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15329 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
965 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,9,6,10,20,11)(3,7,13,22,14,24,30,23)(4,15,18,25,17,26,19,8)(12,27,28,32,16,31,29,21);(1,3)(2,7)(4,16,17,12)(5,13)(6,14)(8,21,25,32)(9,22)(10,24)(11,23)(15,27,26,31)(18,29,19,28)(20,30);(1,4,5,19,6,17,20,18)(2,8,9,26,10,25,11,15)(3,12,13,29,14,16,30,28)(7,21,22,31,24,32,23,27);(1,2)(3,7)(4,8)(5,9)(6,10)(11,20)(12,21)(13,22)(14,24)(15,18)(16,32)(17,25)(19,26)(23,30)(27,28)(29,31)Polycyclic presentation:
2^9|a^2=f,b^2=e,c^2=ef,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,g^b=gi,e^c=ei,f^c=fi,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72280281 |
| maximal normal subgroup of | #72280281 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 332343);