#72250069

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	#24089
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	#6442927	C₂
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#2929 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2519 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)(11,16)(22,27)(23,30)(34,35);(1,2,3,4)(7,15,17,28)(8,19)(9,21)(11,22,16,27)(13,26)(14,25)(18,32,31,29)(20,33)(23,35,30,34)(24,36);(2,4)(5,6)(7,16,17,11)(8,13)(9,14)(10,12)(15,27,28,22)(18,30,31,23)(19,26)(20,24)(21,25)(29,35,32,34)(33,36);(5,7,8,15,9,18,19,29,10,17,20,28,21,31,33,32)(6,11,13,22,14,23,26,34,12,16,24,27,25,30,36,35)

Polycyclic presentation:

2^10|b^2=f,c^2=fh,d^2=g,g^2=i,i^2=j,b^a=bh,c^a=cfh,c^b=ch,d^a=de,d^b=dg,d^c=de,e^c=ej,f^d=fj,g^b=gi,i^b=ij

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250103, #72257102, #72257157
maximal normal subgroup of	#72250103, #72257102, #72257157

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