Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #396371 |
| 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 | C2×#2008 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1423 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,10,7,12,5,11)(3,13,17,24,18,14,16,8)(4,19,21,26,22,9,20,25)(15,28,30,23,31,27,29,32);(1,3,5,16,7,18,6,17)(2,8,10,13,12,24,11,14)(4,15,20,29,22,31,21,30)(9,23,25,27,19,32,26,28);(1,2)(3,14)(4,9)(5,10)(6,11)(7,12)(8,16)(13,18)(15,28)(17,24)(19,22)(20,25)(21,26)(23,29)(27,31)(30,32);(1,4,5,20,7,22,6,21)(2,9,10,25,12,19,11,26)(3,15,16,29,18,31,17,30)(8,23,13,27,24,32,14,28)Polycyclic presentation:
2^9|a^2=f,b^2=e,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,g^b=gi,f^c=fi,g^c=gi,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72213913, #72219486, #72219488, … |
| maximal normal subgroup of | #72213913, #72219486, #72219488, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 303567);