#72307188

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	C₂×#535
commutator, Frattini, cosocle	C₂×C₄²	C₂⁵
socle	C₂²	#34963
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

3075 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^10|a^2=g,b^2=gh,c^2=gh,e^2=hj,f^2=i,g^2=j,h^2=j,b^a=bh,c^a=cf,d^c=dgh,e^a=ehj,e^b=ej,e^c=ei,f^a=fi,f^c=fij,f^d=fj,h^a=hj

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307250, #72307671
maximal normal subgroup of	#72307250, #72307671

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