#72317291

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂³×D₄
commutator	C₂²×C₄	C₂⁴×C₄
Frattini, cosocle	C₂³×C₄	C₂⁵
socle	C₂³	C₂²×C₄×D₄
prime radical	C₂×#3050	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

4783 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^10|a^2=f,b^2=h,c^2=fhij,e^2=fhj,f^2=j,i^2=j,b^a=bg,c^a=chij,c^b=ch,d^a=dh,e^c=ej,i^a=ij

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317307
maximal normal subgroup of	#72317307

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