#72287342

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂×C₄²
commutator	C₂²×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂²×C₄×C₈	C₂³
socle	C₂⁴	C₂×#97
prime radical	C₂⁵	#97
LCS	G ▷ C₂²×C₄ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2073 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^10|b^2=d,c^2=dfij,d^2=h,f^2=i,g^2=j,i^2=j,b^a=bh,c^a=ce,c^b=cegj,d^c=dj,f^b=fj,f^c=fj

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287355, #72287502, #72287569, …
maximal normal subgroup of	#72287355, #72287502, #72287569, …

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