#72237309

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₈² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁵	#95
commutator	C₂²×C₄	C₈²
Frattini, cosocle	C₂²×C₄³	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ C₂²×C₄³ ◁ G
LECS	G ▷ C₂²×C₄³ ▷ C₂⁵ ▷ C₁

2689 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 1024

A minimal faithful permutation representation — 2 generators on 40 points:

(1,2,3,5,4,6,7,8)(17,18)(19,23)(20,22)(21,24)(25,26)(27,31,32,29)(28,30)(33,34)(35,37)(36,38)(39,40);(9,10,11,13,12,14,15,16)(17,19)(18,21,22,23)(20,24)(25,27)(26,29)(28,32)(30,31)(34,37,38,39)

Polycyclic presentation:

2^10|a^2=c,b^2=d,c^2=f,d^2=g,e^2=j,b^a=be,c^b=ci,d^a=dh,e^a=eij,e^b=ehj

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#72237318
Fitting subgroup of	#72237318
maximal subgroup of	#72237318
maximal normal subgroup of	#72237318
maximal quotient of	#72237314

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", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.3);;
gap> SetPower(coll, 2, F.4);;
gap> SetPower(coll, 3, F.6);;
gap> SetPower(coll, 4, F.7);;
gap> SetPower(coll, 5, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);