#72285266

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#1889
commutator	C₂×C₂²⋊C₄	C₂³×C₄
Frattini, cosocle	C₂×#117	C₂⁴
prime radical	C₂×#2021	C₄
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

3267 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^10|b^2=gj,d^2=e,e^2=j,f^2=i,b^a=bg,d^a=dhij,d^b=df,d^c=dj,e^b=ei,f^a=fh,f^b=fhi,g^d=gh,g^f=gj,h^b=hj

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285354, #72285659
maximal normal subgroup of	#72285354, #72285659

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