#72235704

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#1889
commutator	C₂×C₂²⋊C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₂²⋊C₄	C₂⁴
prime radical	#6451874	C₂
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ C₂³×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

3961 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^10|a^2=g,b^2=fg,f^2=i,b^a=bf,c^a=cj,c^b=ce,d^a=di,d^b=di,d^c=dj,e^a=eh,e^b=ehj,f^b=fi,f^c=fhj,f^e=fj,g^b=gi,h^b=hj

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72235725, #72235816, #72235909, …
maximal normal subgroup of	#72235725, #72235816, #72235909, …
maximal quotient of	#72237750

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