#72320145

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂³×D₄
commutator	C₂²×C₄	C₂⁴×C₄
Frattini, cosocle	C₂×C₄²	C₂⁵
socle	C₂³	C₂²×C₄×D₄
prime radical	#36624	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#407 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

4151 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^10|a^2=h,b^2=f,f^2=i,h^2=j,b^a=bhj,e^a=ehj,e^c=eg,e^d=ei,h^b=hj,h^e=hj

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320264
maximal normal subgroup of	#72320264
maximal quotient of	#72321962

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