#72239878

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×C₂²⋊C₄
commutator	C₂²×C₄	C₂⁴×C₄
Frattini, cosocle	C₂×C₄²	C₂⁵
prime radical	C₂⁵	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

4727 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:

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

Polycyclic presentation:

2^10|a^2=f,b^2=fghi,d^2=i,e^2=j,f^2=i,g^2=hj,b^a=bg,c^a=chj,c^b=ci,d^a=di,e^a=ehi,e^b=ej,f^b=fh,g^a=gj,g^b=gh

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239879, #72239903, #72239909
maximal normal subgroup of	#72239879, #72239903, #72239909

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