#72251697

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#2517199
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

4756 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:

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

Polycyclic presentation:

2^10|b^2=f,c^2=fgh,d^2=ij,h^2=j,b^a=bg,c^a=cfi,d^a=dg,d^b=dhj,d^c=dgij,e^c=eghj,e^d=ei,f^d=fj,g^c=gj,h^c=hj,h^e=hj,i^b=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72251715, #72251951, #72251954
maximal normal subgroup of	#72251715, #72251951, #72251954

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