#72322490

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#2177
commutator	C₂×C₁₆	C₂⁴×C₄
Frattini, cosocle	C₄×C₁₆	C₂⁵
socle	C₂²	C₂²×#2177
prime radical	#72322435	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₈² ◁ C₂²×#1311 ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3576 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(18,24)(19,30)(20,23)(21,25)(26,38)(27,46)(28,48)(29,39)(31,32)(33,40)(34,44)(35,36)(37,43)(41,42)(45,47);(1,2)(3,7)(4,8)(5,6)(9,14)(10,12)(11,13)(15,16)(17,18,25,42,19,29,34,48,20,33,43,47,31,27,36,38,22,26,35,46,32,45,37,40,23,28,44,39,30,41,21,24);(2,6)(7,12)(8,13)(14,16)(18,26)(24,38)(27,41)(28,33)(29,45)(39,47)(40,48)(42,46);(1,3,4,9,5,10,11,15)(2,7,8,14,6,12,13,16)(17,19,20,31,22,32,23,30)(18,27,28,29,26,41,33,45)(21,35,34,44,36,25,37,43)(24,39,40,46,38,47,48,42);(18,26)(24,38)(27,41)(28,33)(29,45)(39,47)(40,48)(42,46)

Polycyclic presentation:

2^11|b^2=gi,d^2=f,f^2=h,g^2=ij,i^2=jk,j^2=k,b^a=bg,c^b=ch,d^a=dj,d^b=dj,e^b=ek,f^a=fk,f^b=fk,g^a=gi,g^d=gk,i^a=ij,j^a=jk

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322589, #72322903, #72322953
maximal normal subgroup of	#72322589, #72322903, #72322953

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