#72314431

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	#7622822	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ #20384 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

4803 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^10|b^2=j,c^2=f,e^2=g,g^2=i,i^2=j,b^a=bj,c^b=cj,d^a=dh,d^b=dh,e^a=eh,e^b=eh,e^c=eg,e^d=egij,f^e=fj,g^c=gi,g^d=gij,i^c=ij,i^d=ij

Conjugacy classes & character table

118 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314441, #72315346, #72320956
maximal normal subgroup of	#72314441, #72315346, #72320956

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