#72338492

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#618351
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
prime radical	#6437941	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

4916 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:

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

Polycyclic presentation:

2^10|b^2=i,c^2=h,e^2=ij,g^2=j,b^a=bi,c^a=cf,c^b=cg,d^b=dg,d^c=dghi,e^a=ehj,e^c=egj,e^d=egj,f^d=fj,f^e=fj,g^a=gj,g^c=gj,g^d=gj,h^d=hj,i^c=ij,i^d=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338502
maximal normal subgroup of	#72338502

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