#72256114

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#13956
commutator	C₂³×C₄	C₂³×C₄
Frattini, cosocle	C₂⁴×C₄	C₂⁴
prime radical	#7713911	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

5523 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^10|a^2=h,b^2=g,e^2=ij,b^a=bf,c^a=ch,c^b=ce,d^b=dij,d^c=dhij,e^a=ej,e^b=ei,e^c=eij,e^d=eij,f^a=fij,f^b=fij,f^c=fi,g^a=gij,g^c=gj,h^b=hij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256124, #72256202, #72256412, …
maximal normal subgroup of	#72256124, #72256202, #72256412, …
maximal quotient of	#72266279

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