#72237167

order	2048
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂⁴×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁶
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×#525 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

138888 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 24 points:

(5,6)(7,8)(9,10,13,11)(12,14,16,15)(17,18,20,24)(19,23,21,22);(1,2)(3,4)(18,22)(19,21)(23,24);(1,2)(3,4);(9,11,13,10)(12,15,16,14)(17,19,20,21)(18,23,24,22);(9,12)(10,14)(11,15)(13,16);(2,4)(6,8)(12,16)(14,15)

Polycyclic presentation:

2^11|a^2=g,d^2=g,j^2=k,b^a=bj,d^b=dk,f^a=fi,f^b=fh,f^c=fh,f^e=fgk,j^a=jk,j^b=jk

Conjugacy classes & character table

260 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249436, #72250265, C₂×#72237167, …
maximal normal subgroup of	#72249436, #72250265, C₂×#72237167, …
maximal quotient of	#72257317, #72264672, #72264758, …
automorphism group of	#2184, #2185, #2187, …

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