Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #135513 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #8746 |
| commutator | C25 | C22×C4 |
| Frattini, prime radical, cosocle | C23×D4 | C23 |
| LCS | G ▷ C25 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
528 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 528 conjugacy classes of subgroups — too many to draw. Only its 60 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 — 3 generators on 32 points:
(1,2)(3,11)(4,15,17,8)(5,18,19,9)(6,10)(7,13,21,28)(12,25,27,23)(14,24)(16,22)(20,26)(29,32)(30,31);(1,3,6,14)(2,7,10,21)(4,12,17,27)(5,13,19,28)(8,20,15,29)(9,11,18,24)(16,26,31,32)(22,23,30,25);(1,4,5,16)(2,8,9,22)(3,12,13,26)(6,17,19,31)(7,20,11,23)(10,15,18,30)(14,27,28,32)(21,29,24,25)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=d,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,f^b=fg,e^c=eg,f^c=fh,h^c=hi,f^d=fi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72314438, #72314440, #72314442, … |
| maximal normal subgroup of | #72314438, #72314440, #72314442, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 42710);