#72317083

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁵	C₂⁶
commutator	C₂⁴	C₂⁵×C₄
prime radical	C₂³×#3484	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

712560 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 24 points:

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

Polycyclic presentation:

2^11|d^2=g,f^2=k,b^a=bi,c^a=ci,d^a=dj,d^b=dh,d^c=dh,e^a=ek,f^b=fk,f^d=fk

Conjugacy classes & character table

392 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317136
maximal normal subgroup of	#72317136

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