Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 6 |
| isoclinism | isoclinic class of #94556 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #7349 |
| commutator | C23×C4 | C2×C8 |
| Frattini, cosocle | #1781 | C22 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C23×C4 ▷ C24 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22 ◁ C24 ◁ C25 ◁ #1781 ◁ G | |
| LECS | G ▷ #1781 ▷ C25 ▷ C23 ▷ C22 ▷ C2 ▷ C1 | |
233 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 233 conjugacy classes of subgroups — too many to draw. Only its 24 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 — 2 generators on 32 points:
(1,2,5,9)(3,11,13,26)(4,15,17,30,18,8,19,24)(6,10,20,25)(7,21,22,12,23,28,31,16)(14,27,29,32);(1,3)(2,7,10,23)(4,16)(5,13,20,29)(6,14)(8,11,15,27)(9,22)(12,18)(17,21,19,28)(24,26)(25,31)(30,32)Polycyclic presentation:
2^9|a^2=d,b^2=g,c^2=i,d^2=f,b^a=bc,c^a=ce,e^a=eg,g^a=gh,h^a=hi,c^b=chi,d^b=dei,e^b=ei,f^b=fh,d^c=dg,f^c=fi,e^d=eh,g^d=gi29 conjugacy classes — view the full character table.
| maximal subgroup of | #72237008, #72247768, #72247769, … |
| maximal normal subgroup of | #72237008, #72247768, #72247769, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1752);