#72250418

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#24089
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂×#525	D₄
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#2929 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1937 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:

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

Polycyclic presentation:

2^10|b^2=g,c^2=e,d^2=ej,e^2=i,i^2=j,c^a=ch,c^b=ce,d^a=dghj,d^b=def,e^b=ei,g^c=gj,g^d=gj,h^d=hj,i^b=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250436, #72250632, #72252912
maximal normal subgroup of	#72250436, #72250632, #72252912

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