#72257405

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#32002
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂×#28180	C₂
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#2929 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2140 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)(12,15)(24,25)(26,31)(35,36);(1,2,4,3)(5,6,9,14,10,11,22,23)(7,15,30,26,17,12,18,31)(8,19,20,28,21,13,27,29)(16,25,34,35,32,24,33,36);(2,3)(6,11)(7,16,17,32)(8,20)(9,22)(12,24,15,25)(13,28)(18,34,30,33)(19,29)(21,27)(26,36,31,35);(1,3,4,2)(5,7,8,16,9,18,20,33,10,17,21,32,22,30,27,34)(6,12,19,24,14,26,28,35,11,15,13,25,23,31,29,36)

Polycyclic presentation:

2^10|b^2=fi,c^2=h,d^2=eg,e^2=i,i^2=j,b^a=bf,c^b=cfg,d^a=dfgh,d^b=dg,d^c=de,e^b=ej,e^c=ei,f^d=fj,g^b=gj,h^d=hj,i^c=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257420, #72257453, #72257671
maximal normal subgroup of	#72257420, #72257453, #72257671

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