#72265556

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#16632
commutator, Frattini, cosocle	C₂⁴×C₄	C₂⁴
prime radical	C₂³×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

5985 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:

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

Polycyclic presentation:

2^10|a^2=e,c^2=fghi,d^2=ghi,e^2=j,b^a=beg,c^a=cf,c^b=ch,d^a=dghij,d^c=deghij,e^b=ej,e^c=ej,f^a=fj,f^c=fj,f^d=fj,g^c=gj,h^d=hj,i^d=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265580, #72314015, #72314154
maximal normal subgroup of	#72265580, #72314015, #72314154

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