#72251802

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, socle	C₂	#2924619
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
prime radical	#6437963	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

4724 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^10|c^2=h,d^2=j,h^2=j,b^a=bg,c^a=cf,c^b=cg,d^a=dg,d^b=dgh,d^c=di,e^c=eghij,e^d=eij,f^b=fj,f^d=fj,g^c=gj,h^b=hj,h^d=hj,h^e=hj,i^b=ij,i^c=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72251804, #72251823, #72251956
maximal normal subgroup of	#72251804, #72251823, #72251956

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