#72286147

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#37284
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
socle	C₂	#854634
prime radical	#453994	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

4331 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^10|a^2=g,b^2=j,c^2=h,e^2=h,f^2=j,b^a=bh,d^a=di,d^b=di,d^c=df,e^a=ehi,e^d=eghij,f^d=fj,g^d=gj,g^e=gj,h^d=hj,i^a=ij

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286289, #72339689, #72339724, …
maximal normal subgroup of	#72286289, #72339689, #72339724, …

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