#72313895

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#1438
commutator	C₂³×C₄	C₂³×C₄
Frattini, cosocle	C₂⁴×C₄	C₂⁴
prime radical	#10575805	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

5599 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)(5,13)(6,9)(10,14)(11,16)(12,15)(17,18,21,20)(19,25,26,32)(22,29,30,24)(23,31,28,27);(1,3,6,5)(2,7,9,13)(4,10,12,11)(8,14,15,16)(17,19,21,26)(18,23,20,28)(22,31,30,27)(24,32,29,25);(1,4,6,12)(2,8,9,15)(3,10,5,11)(7,14,13,16)(19,26)(23,28)(25,32)(27,31);(1,5,6,3)(2,7)(4,11,12,10)(8,14)(9,13)(15,16)(17,20,21,18)(19,23)(22,29,30,24)(25,27)(26,28)(31,32)

Polycyclic presentation:

2^10|a^2=j,b^2=ij,c^2=i,d^2=e,g^2=j,b^a=bg,c^b=cj,d^a=dh,d^b=df,e^a=ei,e^b=ej,f^a=fij,f^d=fj,g^a=gj,g^b=gj,h^b=hij,h^d=hi

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313915
maximal normal subgroup of	#72313915

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