#72256579

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁷
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

At least 6086 conjugacy classes of subgroups; only the 42808 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 7 generators on 20 points:

(14,17)(18,20);(10,12);(2,4)(6,8)(10,12)(14,17)(18,20);(10,12)(15,19)(18,20);(6,8)(10,12)(14,17)(18,20);(5,6)(7,8)(9,10,11,12)(13,14)(15,18)(16,17)(19,20);(1,2)(3,4)(13,15,16,19)(14,18,17,20)

Polycyclic presentation:

2^11|f^2=i,g^2=h,f^a=fh,f^b=fi,f^c=fk,f^d=fi,f^e=fk,g^c=gj,g^d=gh

Conjugacy classes & character table

344 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72268601, C₂×#72256579
maximal normal subgroup of	#72268601, C₂×#72256579
maximal quotient of	#120×#2846, C₂×#72256579, #72287919, …

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