#72265185

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#24196
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂×#3155	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #20436 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2192 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^10|a^2=h,b^2=h,c^2=egj,d^2=eghi,e^2=i,f^2=j,g^2=j,i^2=j,c^a=cf,c^b=ce,d^a=dfgij,d^b=de,d^c=di,e^a=ej,e^b=ei,e^c=ej,f^b=fj,f^d=fj,g^b=gj,g^c=gj,g^d=gj,h^c=hj,h^d=hj,i^b=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265241, #72265436, #72306996
maximal normal subgroup of	#72265241, #72265436, #72306996

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