#72226559

order	3200
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Sylow

C₅² ⋊ #3068

Subgroups

	subgroup	G/H
center	C₂	#13106195
commutator, cosocle	C₂×C₄×C₅²	C₂⁴
Frattini	C₂×C₄	C₂²×D₅²
Fitting	C₅²×AGL(1,ℤ/8)	C₂²
socle	C₂×C₅²	C₂×#121
hypercenter	AGL(1,ℤ/8)	D₅²
hypocenter	C₅²	#3068
LCS	G ▷ C₂×C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ AGL(1,ℤ/8)

1230 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 80

A minimal faithful permutation representation — 4 generators on 26 points:

(2,7)(3,10)(4,13)(6,9)(11,12)(14,15)(17,18,20,21,19)(22,23,25,26,24);(2,7)(6,14)(8,16)(9,15)(18,19)(20,21)(23,24)(25,26);(1,3)(2,6)(4,12)(5,11)(7,14)(8,9)(10,13)(15,16);(1,2,4,8,5,7,13,16)(3,9,10,6,11,15,12,14)(23,24)(25,26)

Polycyclic presentation:

2^7,5^2|a^2=f,f^2=g,b^a=bfg,c^a=cg,d^a=de,d^b=dfg,e^a=eg,e^b=eg,f^b=fg,f^d=fg,h^a=h^4,h^c=h^4,i^a=h^3i,i^c=i^4

Conjugacy classes & character table

137 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226564
maximal normal subgroup of	#72226564

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 5, 5]);;
gap> SetPower(coll, 1, F.6);;
gap> SetPower(coll, 6, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.6*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.5);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.7);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.7);;
gap> SetConjugate(coll, 6, 2, F.6*F.7);;
gap> SetConjugate(coll, 6, 4, F.6*F.7);;
gap> SetConjugate(coll, 8, 1, F.8^4);;
gap> SetConjugate(coll, 8, 3, F.8^4);;
gap> SetConjugate(coll, 9, 1, F.8^3*F.9);;
gap> SetConjugate(coll, 9, 3, F.9^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);