#72257751

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×#95
commutator	C₂²×C₄	C₂²×C₄²
Frattini, cosocle	C₂²×C₄²	C₂⁴
prime radical	C₂⁵	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄³ ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

3381 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^10|a^2=e,b^2=g,d^2=j,e^2=h,f^2=ij,b^a=bf,c^a=chj,c^b=cij,d^a=dhj,e^b=ej,f^a=fi,g^a=gij

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257756, #72257764, #72257860
maximal normal subgroup of	#72257756, #72257764, #72257860

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