#72258290

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#9495
commutator	C₂³×C₄	C₂³×C₄
Frattini, cosocle	C₂²×#41	C₂⁴
prime radical	#6437623	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

4770 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:

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

Polycyclic presentation:

2^10|b^2=g,e^2=j,b^a=be,c^a=cf,c^b=cfj,d^a=dhj,d^b=dhj,e^a=ej,e^b=eh,e^c=ei,f^b=fi,f^d=fj,g^a=ghj,g^c=gi,g^f=gj,h^c=hj,i^b=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72258313, #72258396, #72258446
maximal normal subgroup of	#72258313, #72258396, #72258446

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