#72258939

order	2048
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#14335
commutator	C₂⁶	C₂³×C₄
Frattini, cosocle	C₂⁷	C₂⁴
prime radical	#72225665	C₂
LCS	G ▷ C₂⁶ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁷ ◁ G
LECS	G ▷ C₂⁷ ▷ C₂³ ▷ C₁

At least 6155 conjugacy classes of subgroups; only the 480 normal subgroups are shown.

Full subgroup data →

Presentation

Polycyclic presentation:

2^11|a^2=g,b^a=bh,c^a=ce,c^b=cef,d^a=df,d^b=def,e^a=ei,e^b=ek,f^a=fj,f^b=fk,g^c=gi,g^d=gj,h^c=hijk,h^d=hijk

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72258971, #72266247, #72266248, …
maximal normal subgroup of	#72258971, #72266247, #72266248, …
maximal quotient of	#72317153, #72317154

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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.7);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.5);;
gap> SetConjugate(coll, 3, 2, F.3*F.5*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.5*F.6);;
gap> SetConjugate(coll, 5, 1, F.5*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.11);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.11);;
gap> SetConjugate(coll, 7, 3, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.9*F.10*F.11);;
gap> SetConjugate(coll, 8, 4, F.8*F.9*F.10*F.11);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);