#72252656

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#24128
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	#6442896	C₂
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#2929 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2182 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:

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

Polycyclic presentation:

2^10|a^2=f,c^2=fgij,g^2=j,h^2=i,i^2=j,b^a=bfgj,c^a=cfghj,c^b=cfghj,d^b=dgi,d^c=de,e^b=ej,f^c=fj,g^a=gj,g^b=gj,h^a=hi,h^b=hij,h^c=hij,i^a=ij,i^b=ij,i^c=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72252658, #72252688, #72252924
maximal normal subgroup of	#72252658, #72252688, #72252924

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