Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12148 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | B(4,𝔽2) |
| commutator | C2×C22⋊C4 | C22×C4 |
| Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | C2×#1595 | C2 |
| LCS | G ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24 ◁ C22×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1053 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 1053 conjugacy classes of subgroups — too many to draw. Only its 88 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(2,4)(5,6,8,10,9,7,12,11)(13,14,17,19)(15,18)(16,20);(1,2,3,4)(5,7)(6,9)(8,10)(11,12)(13,15)(14,18,19,20)(16,17);(5,6)(7,9)(8,10)(11,12)(14,18)(15,16)(19,20)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=gi,e^2=g,f^2=gi,b^a=bd,c^a=ce,d^a=di,e^a=egh,h^a=hi,c^b=ci,d^b=dg,e^b=ehi,f^b=fg,d^c=dh,e^c=eg,f^c=fh,e^d=ei,f^e=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72282818, #72283102, #72283154 |
| maximal normal subgroup of | #72282818, #72283102, #72283154 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44561);