#72257302

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#32468
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂⁵	#137
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #20385 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1352 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:

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

Polycyclic presentation:

2^10|a^2=e,c^2=h,d^2=fgh,e^2=i,f^2=j,i^2=j,b^a=bg,c^a=cefj,d^a=dfgj,d^b=dh,d^c=dfhij,e^c=ei,e^d=ej,f^c=fj,f^d=fj,g^d=gj,h^a=hj,i^c=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257311, #72257342, #72257680
maximal normal subgroup of	#72257311, #72257342, #72257680

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