#72328751

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁵	C₂⁵
prime radical	C₂³×#120	C₂²
LCS	G ▷ C₂⁵ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

At least 6263 conjugacy classes of subgroups; only the 959 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^10|a^2=j,b^2=h,c^2=j,d^2=i,e^2=g,b^a=bg,c^a=cj,c^b=cf,d^a=dh,d^b=dhi,d^c=dhij,e^a=efhi,e^b=efh,e^c=efghij,e^d=egi

Conjugacy classes & character table

118 conjugacy classes — view the full character table.

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