#72323769

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁵	C₂⁶
commutator	C₂⁴	C₂⁵×C₄
prime radical	C₂²×#15258	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

At least 7108 conjugacy classes of subgroups; only the 8344 normal subgroups are shown.

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:

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

Polycyclic presentation:

2^11|b^2=gk,e^2=hj,f^2=ghj,b^a=bg,e^c=ej,e^d=eh,f^b=fghj,f^c=fi,f^d=fghij

Conjugacy classes & character table

296 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72323772
maximal normal subgroup of	#72323772

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", "k");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 2, F.7*F.11);;
gap> SetPower(coll, 5, F.8*F.10);;
gap> SetPower(coll, 6, F.7*F.8*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.7*F.8*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.7*F.8*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);