#72337431

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#3027
commutator, Frattini, cosocle	C₂²×#41	C₂⁴
socle	C₂²	#32926
LCS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

5064 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2)(3,4)(6,12)(7,17)(13,29)(15,27)(16,22)(18,35)(20,34)(21,25);(6,13,16,15)(7,18,21,20)(12,27,22,29)(17,34,25,35);(1,2)(3,4)(5,6,8,22)(7,19,25,33)(9,12,11,16)(10,13,23,27)(14,18,28,34)(15,24,29,26)(17,30,21,36)(20,31,35,32);(2,4)(5,7)(6,14)(8,17)(9,25)(10,18)(11,21)(12,28)(13,30)(15,33)(16,32)(19,27)(20,26)(22,31)(23,35)(24,34)(29,36)

Polycyclic presentation:

2^10|b^2=i,c^2=fj,g^2=j,h^2=j,b^a=bi,c^a=cf,c^b=ch,d^a=de,d^b=dhij,d^c=degj,e^b=ei,e^c=ej,f^b=fi,f^d=fj,g^c=gj,g^d=gj,h^a=hi,h^c=hij,h^e=hj,h^f=hj,i^c=ij,i^d=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72337432, #72337449, #72339313
maximal normal subgroup of	#72337432, #72337449, #72339313

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