#72231746

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#23900
commutator, Frattini, cosocle	C₂×#41	C₂⁵
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

16069 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:

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

Polycyclic presentation:

2^10|b^2=h,e^2=fgj,f^2=j,b^a=bh,e^a=ef,e^b=eg,e^c=efgj,e^d=ej,f^a=fj,f^b=fi,f^c=fj,g^b=gi,h^e=hi,h^f=hj,h^g=hj,i^a=ij,i^b=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237558, #72240979, #72241001, …
maximal normal subgroup of	#72237558, #72258280, C₂×#72231746, …
maximal quotient of	C₂×#72231746, #72280991, #72281052, …
automorphism group of	#1391, #1392

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