#72234199

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#392109
commutator	C₂⁵	C₂³×C₄
Frattini, cosocle	C₂³×D₄	C₂⁴
prime radical	#10586066	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ C₂×H(7,𝔽₂) ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂² ▷ C₂ ▷ C₁

31127 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)(4,8)(5,9)(6,10)(7,14)(12,18)(13,24)(15,21)(16,22)(17,23)(19,27)(20,25)(26,29)(28,31)(30,32);(1,3,6,14)(2,7,10,11)(4,12,15,25)(5,13,17,27)(8,20,21,18)(9,19,23,24)(16,26,28,32)(22,30,31,29);(4,15)(8,21)(12,25)(18,20);(1,4,5,16,6,15,17,28)(2,8,9,22,10,21,23,31)(3,12,13,26,14,25,27,32)(7,18,19,29,11,20,24,30)

Polycyclic presentation:

2^10|b^2=j,d^2=e,e^2=j,b^a=bj,d^a=dg,d^b=dh,d^c=df,e^c=ei,f^d=fi,f^e=fj,g^b=gj,h^a=hj,i^d=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72260689, #72260919
maximal normal subgroup of	#72260689, #72260919
maximal quotient of	#72234200

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