#72239737

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#13967
commutator	C₂×C₄²	C₂³×C₄
Frattini, cosocle	C₂²×C₄²	C₂⁴
prime radical	#6453969	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

3945 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^10|a^2=h,b^2=ij,d^2=g,e^2=i,h^2=j,b^a=bhj,c^a=cf,c^b=cfi,d^a=dij,d^b=dij,d^c=de,e^a=ej,e^b=ej,e^c=ei,f^d=fj,g^c=gi,h^b=hj

Conjugacy classes & character table

97 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239784
maximal normal subgroup of	#72239784

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