#72285605

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#46996
commutator, Frattini, cosocle	C₂⁵	C₂⁵
prime radical	#454367	C₂
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

5248 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^10|c^2=fg,d^2=ij,e^2=j,b^a=bi,c^a=cf,c^b=ch,d^a=di,d^b=dh,d^c=dfj,e^b=eghij,e^d=efgj,f^b=fj,f^d=fj,g^d=gj,h^c=hj,i^c=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285704
maximal normal subgroup of	#72285704

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