#72250447

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#32554
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂²×#509	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #20436 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1789 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:

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

Polycyclic presentation:

2^10|b^2=h,c^2=fgi,d^2=efh,e^2=ij,i^2=j,c^a=cg,c^b=cf,d^a=de,d^b=defi,d^c=dij,e^a=ei,e^b=eij,e^d=ej,f^c=fj,g^c=gj,g^d=gj,h^d=hj,i^a=ij,i^b=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250490, #72250807, #72257241, …
maximal normal subgroup of	#72250490, #72250807, #72257241, …

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