#72234060

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#10745
commutator	C₂×C₄⋊C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄×D₄	C₂³
prime radical	C₂×#2635	C₂²
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

1999 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^10|a^2=g,b^2=e,c^2=degh,d^2=j,f^2=i,b^a=bf,c^a=cf,c^b=cd,d^a=dh,d^b=dh,d^c=dhi,e^c=ehj,e^d=ej,f^b=fi,f^c=fhi,f^d=fj,g^b=gi,g^c=gi,h^b=hj,h^c=hj

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234066, #72234082, #72234090
maximal normal subgroup of	#72234066, #72234082, #72234090

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