#72238947

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁵
socle	C₂²	C₂²×#399
prime radical	C₂³×#41	Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

4501 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

(17,18,20,24,21,23,29,31)(19,25,27,30,26,32,28,22);(1,2)(3,9)(4,12)(5,7)(6,11)(8,13)(10,15)(14,16)(22,30)(25,32);(1,3,4,11,5,10,13,16)(2,6,8,9,7,14,12,15)(17,19,20,28,21,26,29,27)(18,22,31,25,23,30,24,32);(2,7)(3,10)(8,12)(11,16);(2,7)(6,14)(8,12)(9,15)

Polycyclic presentation:

2^10|a^2=h,c^2=fgh,f^2=i,h^2=j,b^a=bg,c^a=cghj,c^b=cf,d^b=di,d^c=di,e^b=ei,f^a=fj,f^b=fi,g^c=gj,h^c=hj

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238977, #72239012, #72239164
maximal normal subgroup of	#72238977, #72239012, #72239164

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