#72317218

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×#106
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂⁶	SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁵×C₄ ◁ C₂⁵×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3945 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^10|a^2=e,c^2=ghij,h^2=i,i^2=j,b^a=bg,c^a=cghij,c^b=cf,d^c=dg,e^c=ej,h^a=hi,i^a=ij

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317220, #72317221, #72317254
maximal normal subgroup of	#72317220, #72317221, #72317254

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