#72322447

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	#32002
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂×#3067	C₂²
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #20383 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1595 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)(6,13,16,30)(7,19)(9,11)(10,25)(14,32,20,28)(15,29,33,36)(17,31,18,26)(21,35)(23,34)(24,27);(5,6)(7,20)(8,15)(9,18)(10,26)(11,17)(12,16)(13,24)(14,19)(21,29)(22,33)(23,28)(25,30)(27,31)(32,35)(34,36);(1,2,3,4)(5,7,9,22,12,19,11,8)(6,14,18,15,16,20,17,33)(10,21,24,35,27,34,25,23)(13,28,31,29,30,32,26,36);(5,8,11,19,12,22,9,7)(6,15,18,20,16,33,17,14)(10,23,25,34,27,35,24,21)(13,29,31,32,30,36,26,28)

Polycyclic presentation:

2^10|a^2=e,c^2=efg,d^2=i,g^2=j,h^2=i,i^2=j,b^a=bh,c^a=cf,c^b=cj,d^a=dgij,d^b=dgij,d^c=de,e^b=ej,f^c=fj,f^d=fj,g^a=gj,g^b=gj,g^c=gj,g^d=gj,h^a=hi,h^b=hij,i^a=ij,i^b=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322466, #72322591
maximal normal subgroup of	#72322466, #72322591

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