#72231459

order	2048
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×#442
commutator	C₂²×C₈	C₂⁴×C₄
Frattini, cosocle	C₂³×C₈	C₂⁵
prime radical	#72231322	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁵×C₄ ◁ C₂⁴×#97 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

83361 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 24 points:

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

Polycyclic presentation:

2^11|d^2=f,e^2=ghj,h^2=jk,j^2=k,e^a=eg,e^b=ei,e^c=eh,e^d=eghj,f^e=fk,h^c=hj,h^d=hjk,j^c=jk,j^d=jk

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256975, #72257753, #72307508, …
maximal normal subgroup of	#72256975, #72257753, #72307508, …
maximal quotient of	#72286863, C₂×#72231459, #72318716
automorphism group of	#1380, #2176, #2177, …

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