#72314425

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	#7623308	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ #20384 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3009 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^10|b^2=j,c^2=f,e^2=ghj,g^2=i,i^2=j,b^a=bj,c^b=cj,d^a=dh,d^b=dh,e^a=ehj,e^c=eg,e^d=egi,f^e=fj,g^c=gi,g^d=gij,i^c=ij,i^d=ij

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314441, #72314475, #72315218
maximal normal subgroup of	#72314441, #72314475, #72315218

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