Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #142220 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×C4⋊C4 | C23 |
| prime radical | C24 | #120 |
| LCS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C22×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
589 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 589 conjugacy classes of subgroups — too many to draw. Only its 80 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,11,8)(7,14,17,20)(9,19,10,13)(12,16,18,15)(21,22,26,24)(23,29,32,36)(25,28,34,33)(27,35,30,31);(5,7,11,17)(6,12,8,18)(9,16,10,15)(13,20,19,14)(21,23)(22,27)(24,30)(25,31)(26,32)(28,29)(33,36)(34,35);(2,4)(5,8,11,6)(7,14,17,20)(9,19,10,13)(12,15,18,16)(21,24,26,22)(23,30,32,27)(25,28,34,33)(29,31,36,35)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=gi,d^2=g,f^2=g,b^a=bd,c^a=ce,d^a=dg,f^a=fhi,c^b=cf,d^b=dg,f^b=fg,h^b=hi,d^c=dh,f^c=fg,f^d=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72227643 |
| maximal normal subgroup of | #72227643 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 50088);