Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #444648 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15784 |
| 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 | |
1093 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10,6,9,20,25)(3,11,13,27,14,7,19,22)(4,15,18,31,17,8,23,26)(12,21,16,24,29,28,32,30);(1,3)(2,7)(4,16)(5,19)(6,14)(8,24)(9,11)(10,27)(12,18)(13,20)(15,30)(17,32)(21,26)(22,25)(23,29)(28,31);(1,4,5,18,6,17,20,23)(2,8,25,31,9,15,10,26)(3,12,13,16,14,29,19,32)(7,21,27,30,11,28,22,24);(1,3)(2,7,9,11)(4,12)(5,13)(6,14)(8,21,15,28)(10,22,25,27)(16,18)(17,29)(19,20)(23,32)(24,31,30,26)Polycyclic presentation:
2^9|a^2=g,c^2=eg,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,g^b=gi,h^b=hi,f^c=fi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72338914, #72338917, #72338927 |
| maximal normal subgroup of | #72338914, #72338917, #72338927 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 351844);