Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #423419 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15314 |
| 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 | |
1304 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)(3,8)(4,16,17,27)(5,11)(6,21)(7,12)(9,18,26,20)(10,14)(13,28,29,25)(15,23)(19,24)(22,30,32,31);(1,3)(2,8)(4,13,17,29)(5,19)(6,14)(7,15)(9,22,26,32)(10,21)(11,24)(12,23)(16,25,27,28)(18,31,20,30);(1,4,5,20,7,17,6,18)(2,9,11,27,12,26,21,16)(3,13,14,31,15,29,19,30)(8,22,10,25,23,32,24,28);(1,3)(2,10)(4,13)(5,14)(6,19)(7,15)(8,21)(9,25)(11,23)(12,24)(16,22)(17,29)(18,30)(20,31)(26,28)(27,32)Polycyclic presentation:
2^9|a^2=e,b^2=e,c^2=ef,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72338910, #72338914, #72338919, … |
| maximal normal subgroup of | #72338910, #72338914, #72338919, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330615);