#72266710

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#23984
commutator, Frattini, cosocle	C₂³×D₄	C₂⁴
prime radical	#347042	C₂
LCS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#120 ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

3306 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^10|a^2=g,b^a=be,c^a=cj,c^b=cf,d^c=dg,e^a=ei,e^c=eh,f^a=fhj,f^d=fi,f^e=fj,g^b=gi,h^b=hj

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72266711, #72266715, #72266717
maximal normal subgroup of	#72266711, #72266715, #72266717
maximal quotient of	#72337753

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