Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #473877 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16885 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
897 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,5,12,7,9)(3,13,17,22,14,8,16,24)(4,18,21,10,19,25,20,26)(15,29,30,23,28,32,31,27);(1,3,6,16,5,14,7,17)(2,8,9,24,12,13,11,22)(4,15,20,30,19,28,21,31)(10,23,25,29,26,27,18,32);(2,9)(3,14)(4,19)(6,7)(8,22)(10,25)(11,12)(13,24)(15,28)(18,26)(23,29)(27,32);(1,4)(2,10)(3,15)(5,19)(6,20)(7,21)(8,23)(9,25)(11,18)(12,26)(13,27)(14,28)(16,30)(17,31)(22,32)(24,29)Polycyclic presentation:
2^9|a^2=fg,b^2=egh,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,g^b=gi,d^c=de,f^c=fi,h^c=hi,g^d=gi,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72338917, #72338920, #72338927, … |
| maximal normal subgroup of | #72338917, #72338920, #72338927, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 381073);