#72257153

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#31680
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂×#24281	C₂
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#2929 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3786 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:

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

Polycyclic presentation:

2^10|a^2=f,d^2=efh,e^2=ij,f^2=j,i^2=j,b^a=bh,c^a=cfhij,d^a=de,d^b=dg,d^c=defgj,e^a=ei,e^c=ei,e^d=ej,f^c=fj,g^a=gj,h^d=hj,i^a=ij,i^c=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257162, #72257230, #72257738
maximal normal subgroup of	#72257162, #72257230, #72257738

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