#72307201

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#535
commutator, Frattini, cosocle	C₂×C₄²	C₂⁵
socle	C₂²	#34963
prime radical	#34547	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2479 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)(3,8)(4,7)(5,6)(9,10,13,15,14,19,26,18)(11,16,22,28,23,31,21,29)(12,24,27,33,25,32,38,17)(20,34,37,40,35,39,36,30);(2,6)(7,8)(10,15,19,18)(12,25)(16,28,31,29)(17,32,33,24)(20,35)(27,38)(30,39,40,34)(36,37);(2,7)(3,4)(6,8)(10,15,19,18)(12,25)(16,29,31,28)(17,32,33,24)(20,35)(27,38)(30,34,40,39)(36,37);(9,11)(10,16)(12,20)(13,22)(14,23)(15,28)(17,30)(18,29)(19,31)(21,26)(24,34)(25,35)(27,37)(32,39)(33,40)(36,38);(1,3,5,4)(2,7,6,8)(9,12,26,38,14,25,13,27)(10,17,15,24,19,33,18,32)(11,20,21,36,23,35,22,37)(16,30,28,34,31,40,29,39)

Polycyclic presentation:

2^10|a^2=g,b^2=ghi,c^2=ghi,e^2=hj,f^2=i,g^2=j,h^2=j,b^a=bh,c^a=cf,d^c=dghi,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, #72307699
maximal normal subgroup of	#72307250, #72307699

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*F.9);;
gap> SetPower(coll, 3, F.7*F.8*F.9);;
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*F.9);;
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);