#72239388

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄³ · Schur multiplier (H₂) = C₂⁵×C₄²×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂×C₄²	C₂⁵
commutator	C₂²	C₂²×C₄³
socle	C₂³	C₂³×C₄²
prime radical	C₂⁵	C₂×C₄²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

5727 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^10|a^2=g,b^2=f,d^2=i,e^2=h,f^2=j,g^2=i,c^a=cij,c^b=cj,d^a=di,d^b=dij,e^a=ej

Conjugacy classes & character table

304 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72239403
maximal subgroup of	#72239390, #72239403
maximal normal subgroup of	#72239390, #72239403

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