#72227779

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁵	C₂⁵
prime radical	C₂⁸	C₂²
LCS	G ▷ C₂⁵ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

115045 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^10|b^2=h,c^2=h,d^2=i,b^a=bj,c^a=cg,c^b=cf,d^a=dg,d^b=di,d^c=dhi,e^b=egj,e^c=ej,e^d=ej

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#72227820, #72227822
maximal subgroup of	#72227783, #72227820
maximal normal subgroup of	#72227783, #72227820

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