#72227993

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#1560
commutator	C₂×C₄²	C₂×C₄²
Frattini, cosocle	C₂×C₄³	C₂³
socle	C₂²	#7631
prime radical	C₂⁵	#99
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×C₄³ ◁ G
LECS	G ▷ C₂×C₄³ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2275 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 20 points:

(5,6)(7,14,16,20,17,19,15,10)(8,12)(9,13)(11,18);(1,2,3,4)(5,7)(6,10)(8,16)(9,17)(11,19)(12,14)(13,20)(15,18);(7,15,17,16)(10,14,20,19)

Polycyclic presentation:

2^10|a^2=d,b^2=f,c^2=i,d^2=i,e^2=hj,g^2=j,b^a=be,c^a=ci,c^b=cgh,d^b=dg,e^a=egh,e^b=ehj,g^b=gj,h^a=hj,i^b=ij

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227994
maximal normal subgroup of	#72227994
maximal quotient of	#72268639

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