#72258509

order	3200
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁹×C₅

Aliases

No aliases for this group.

Splittings

Fitting	C₅²×#379 ⋊ C₂
Sylow	C₅² ⋊ #2846

Subgroups

	subgroup	G/H
center, Frattini	C₂³	C₂³×C₅⋊D₅
commutator, socle, cosocle	C₂³×C₅²	C₂⁴
Fitting	C₅²×#379	C₂
hypercenter	#379	C₅⋊D₅
hypocenter	C₅²	#2846
LCS	G ▷ C₂³×C₅² ▷ C₅²
UCS	C₁ ◁ C₂³ ◁ #379

13688 conjugacy classes of subgroups; only the 635 normal subgroups are shown.

normal subgroups only

This group has 13688 conjugacy classes of subgroups — too many to draw. Only its 635 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

transitive degree

μ_t(G) = 400

Polycyclic presentation:

2^7,5^2|a^2=g,b^a=be,c^a=cf,d^a=dg,h^d=h^4,i^d=i^4

Conjugacy classes & character table

380 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₅⋊D₅×#2846
maximal normal subgroup of	C₅⋊D₅×#2846

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.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 8, 4, F.8^4);;
gap> SetConjugate(coll, 9, 4, F.9^4);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);