#72342998

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄²
Frattini, cosocle	C₂²×C₄²	C₂⁴
socle	C₂²	C₂²⋊C₄×C₄⋊C₄
prime radical	C₂²×#41	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

2073 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^10|a^2=e,b^2=eij,c^2=eg,d^2=i,e^2=i,h^2=j,b^a=bh,c^b=cj,d^a=di,d^b=di,d^c=df,f^c=fi,g^d=gi,h^a=hj,h^b=hj

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343013, #72343096, #72343234
maximal normal subgroup of	#72343013, #72343096, #72343234

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