#72285391

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

3304 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285434, #72285471, #72285659
maximal normal subgroup of	#72285434, #72285471, #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);;
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, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
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);