Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #432844 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15467 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1627 conjugacy classes of subgroups; only the 188 normal subgroups are shown.
This group has 1627 conjugacy classes of subgroups — too many to draw. Only its 188 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,7,12,22)(4,14)(5,18)(6,9)(8,17)(10,20)(11,28,15,31)(13,26,19,24)(16,27)(21,25)(23,30,32,29);(1,3)(2,7,9,22)(4,15,16,11)(5,19)(6,12)(8,23)(10,26,18,24)(13,20)(14,31)(17,29,21,30)(25,32)(27,28);(5,20)(7,22)(10,18)(13,19)(14,27)(17,21)(23,32)(29,30);(1,4)(2,8)(3,11)(5,21)(6,16)(7,23)(9,25)(10,27)(12,15)(13,30)(14,18)(17,20)(19,29)(22,32)(24,31)(26,28)Polycyclic presentation:
2^9|a^2=eg,b^2=ef,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,h^c=hi,f^d=fi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72284248, #72284342, #72284423, … |
| maximal normal subgroup of | #72284248, #72284342, #72284423, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 340040);