#72340070

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁵	C₂⁶
commutator	C₂⁴	C₂⁵×C₄
prime radical	C₂⁵×#121	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

831792 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:

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

Polycyclic presentation:

2^11|b^2=g,c^2=k,d^2=gi,b^a=bh,c^b=cj,d^a=dhi,d^c=djk,e^b=ehj,e^d=ehj,f^b=fik,f^d=fik

Conjugacy classes & character table

368 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340079
maximal normal subgroup of	#72340079

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