#72335529

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

At least 6759 conjugacy classes of subgroups; only the 3385 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 28 points:

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

Polycyclic presentation:

2^10|b^2=h,e^2=ghj,f^2=gi,b^a=bg,c^b=chi,d^b=dghj,e^a=ej,e^b=egj,e^c=eh,e^d=eghj,f^d=fgi

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72335547
maximal normal subgroup of	#72335547
maximal quotient of	#72336103

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