#72239271

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁶
commutator, socle	C₂³	C₂⁵×C₄
prime radical	#7627910	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

16725 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^10|b^2=g,c^2=h,f^2=i,i^2=j,b^a=bg,d^b=dh,e^d=ej,f^c=fj

Conjugacy classes & character table

244 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239291, #72257040, #72257235, …
maximal normal subgroup of	#72239291, #72257040, #72257235, …
maximal quotient of	#72239441

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