#72239663

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂⁴×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁶
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

24121 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 28 points:

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

Polycyclic presentation:

2^10|d^2=g,e^2=h,f^2=i,i^2=j,d^a=dg,e^c=ej,f^a=fi,f^b=fj,f^d=fh,i^a=ij

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72251046, #72265167
maximal normal subgroup of	#72251046, #72265167
maximal quotient of	#72239667

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