#72339682

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	#1574561
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

5892 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)(3,11,14,7)(4,8)(5,10)(6,9)(12,21,28,25)(13,26,30,22)(15,31,29,23)(16,18)(17,24)(19,20)(27,32);(1,3)(2,7)(4,15,17,29)(5,13)(6,14)(8,23,24,31)(9,11)(10,22)(12,27,28,18)(16,25,32,21)(19,26)(20,30);(1,4)(2,8,9,24)(3,12)(5,18)(6,17)(7,21,11,25)(10,16,19,32)(13,29)(14,28)(15,30)(20,27)(22,23,26,31);(2,9)(7,11)(8,24)(10,19)(12,28)(15,29)(16,32)(22,26);(1,2)(3,7)(4,16)(5,19)(6,9)(8,18)(10,20)(11,14)(12,23)(13,26)(15,25)(17,32)(21,29)(22,30)(24,27)(28,31)

Polycyclic presentation:

2^10|a^2=i,b^2=f,c^2=g,h^2=j,b^a=bg,c^a=cfij,c^b=cfgh,d^a=dj,d^b=df,d^c=di,e^a=ei,e^c=ehj,e^d=ej,f^c=fj,g^a=gj,g^e=gj,h^e=hj,i^b=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72339689
maximal normal subgroup of	#72339689

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