#72250014

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#24248
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #20383 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2954 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:

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

Polycyclic presentation:

2^10|a^2=f,c^2=g,d^2=h,g^2=i,i^2=j,c^a=cg,c^b=cgij,d^a=df,d^b=dh,d^c=de,e^a=ej,f^c=fj,g^a=gi,g^b=gij,i^a=ij,i^b=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250023, #72256936, #72257284
maximal normal subgroup of	#72250023, #72256936, #72257284

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