#72250004

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	#24128
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂×#13318	C₂
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #20383 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2350 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:

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

Polycyclic presentation:

2^10|a^2=f,b^2=f,c^2=g,d^2=h,g^2=i,i^2=j,c^a=ci,c^b=cg,d^a=dfh,d^b=dh,d^c=de,e^a=ej,f^c=fj,g^a=gj,g^b=gi,i^b=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250023, #72250151, #72257131
maximal normal subgroup of	#72250023, #72250151, #72257131
maximal quotient of	#72308767

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