#72231985

order	2048
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#1368
commutator	C₂×C₁₆	C₂³×C₈
Frattini, cosocle	#1311	C₂⁴
prime radical	C₂×#41	#365
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₈×#131 ◁ #479943 ◁ G
LECS	G ▷ #1311 ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

789 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 1024

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2,4,9,5,10,14,22,6,8,15,20,16,21,26,30)(3,7,12,17,13,18,25,27,11,19,23,28,24,29,31,32)(34,39,41,50,40,49,53,52)(35,43,44,57)(36,46)(37,48)(42,56,55,51,54,64,63,62)(45,60,59,58)(47,61);(1,3)(2,7)(4,12)(5,13)(6,11)(8,19)(9,17)(10,18)(14,25)(15,23)(16,24)(20,28)(21,29)(22,27)(26,31)(30,32)(34,40)(39,49)(41,53)(42,54)(50,52)(51,62)(55,63)(56,64);(2,8)(3,11)(9,20)(10,21)(12,23)(13,24)(22,30)(25,31)(34,40)(39,49)(41,53)(42,54)(50,52)(51,62)(55,63)(56,64);(33,34,35,39,36,42,43,51,37,41,45,52,46,55,58,56,38,40,44,49,47,54,57,62,48,53,59,50,61,63,60,64)

Polycyclic presentation:

2^11|a^2=e,d^2=f,e^2=g,f^2=h,g^2=i,h^2=j,j^2=k,c^a=ci,c^b=ci,d^a=df,d^b=dk,d^c=dk,e^d=ejk,f^a=fh,f^e=fk,g^d=gk,h^a=hj,j^a=jk

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72231993
maximal normal subgroup of	#72231993

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