#72225673

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#379
commutator	C₂⁵	C₂³×C₄
Frattini, cosocle	C₂⁶	C₂⁴
prime radical	#7630939	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁷ ◁ G
LECS	G ▷ C₂⁶ ▷ C₂² ▷ C₁

22173 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 20 points:

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

Polycyclic presentation:

2^10|b^2=i,c^2=eg,d^2=eg,b^a=bj,c^a=ch,c^b=ce,d^a=dh,d^b=def,e^a=ej,e^c=ei,e^d=ei,g^b=gi,g^c=gi,g^d=gi,h^b=hj,h^c=hj,h^d=hj

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225721, #72258103, #72258111, …
maximal normal subgroup of	#72225721, #72258103, #72258111, …
maximal quotient of	#72225676, #72235061, #72236520, …

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