#72341213

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#20609
commutator	C₂×C₂²⋊C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₂²⋊C₄	C₂⁴
prime radical	#7806788	C₂
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂×#518 ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

3906 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^10|b^2=f,d^2=j,g^2=hij,b^a=be,c^b=ceh,d^a=dj,d^b=dfghij,d^c=di,e^d=ei,f^c=fj,f^d=fh,g^a=gij,g^b=gi,g^c=gij,g^d=gi,g^e=gj,g^f=gj,h^b=hj,i^b=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341243
maximal normal subgroup of	#72341243

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