#72250168

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#23573
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂×#13334	C₂
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #20436 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2402 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^10|a^2=f,c^2=fhj,d^2=g,e^2=i,g^2=i,i^2=j,b^a=bh,c^a=cfj,c^b=cfhj,d^a=de,d^b=deij,d^c=degij,e^a=ei,e^b=eij,e^c=ej,f^d=fj,g^a=gi,g^b=gij,i^a=ij,i^b=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250221, #72250235, #72256950
maximal normal subgroup of	#72250221, #72250235, #72256950

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