Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #142216 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | C2×#534 | C22 |
| LCS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
915 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 915 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) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(5,6,9,7)(8,10,12,11)(13,14,17,20)(15,21,23,28)(16,24,26,19)(18,25,27,22);(6,10)(7,11)(8,12)(13,15)(14,18)(16,25)(17,23)(19,28)(20,27)(21,24)(22,26);(2,4)(5,7,9,6)(8,10,12,11)(16,26)(19,24)Polycyclic presentation:
2^9|a^2=g,c^2=gi,d^2=g,f^2=gi,b^a=bd,c^a=ce,d^a=dg,f^a=fhi,c^b=cf,d^b=dg,f^b=fgi,h^b=hi,d^c=dh,f^c=fgi,f^d=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72283843, #72284405, #72284464, … |
| maximal normal subgroup of | #72283843, #72284405, #72284464, … |
| maximal quotient of | #72285607, #72285783, #72292071 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 50079);