#72339977

order	2048
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁵	C₂⁶
prime radical	C₂⁵×#120	C₂
LCS	G ▷ C₂⁵ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

2680048 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^11|b^2=gh,c^2=ghjk,d^2=ghi,b^a=bg,c^a=ch,c^b=cghj,d^a=dh,d^b=dghi,d^c=dijk,e^b=ehijk,e^c=ehijk,e^d=ehijk,f^b=fghk,f^c=fghk,f^d=fghk

Conjugacy classes & character table

320 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72339997
maximal normal subgroup of	#72339997

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