Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #485637 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×C4⋊C4 | C24 |
| prime radical | #23587 | C2 |
| LCS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
873 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 873 conjugacy classes of subgroups — too many to draw. Only its 122 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) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,10)(4,8)(5,14)(6,9)(7,12)(11,15)(13,16)(17,18)(19,22)(20,23)(21,25)(24,29)(26,30)(27,32)(28,31);(1,3,6,13)(2,7,9,15)(4,12,5,11)(8,16,14,10)(17,19,21,27)(18,22,25,32)(20,28,29,26)(23,30,24,31);(18,23)(20,29)(22,30)(24,25)(26,28)(31,32);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,10,15,16)Polycyclic presentation:
2^9|b^2=gi,d^2=g,e^2=g,f^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,h^a=hi,d^b=dg,e^b=eg,f^b=fhi,e^c=eh,f^c=fi,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72338532, #72338703, #72338759 |
| maximal normal subgroup of | #72338532, #72338703, #72338759 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 393158);