#72303243

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#1438
commutator	C₂³×C₄	C₂³×C₄
Frattini, cosocle	C₂⁴×C₄	C₂⁴
socle	C₂²	C₂×#1438
prime radical	#10575874	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

5895 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^10|a^2=i,b^2=h,c^2=e,d^2=i,h^2=j,b^a=bh,c^a=cf,c^b=cg,e^b=ei,f^b=fij,g^a=gij,g^c=gi,h^a=hj

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303262, #72303319
maximal normal subgroup of	#72303262, #72303319

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