#72232423

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×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 6027 conjugacy classes of subgroups; only the 3341 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:

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

Polycyclic presentation:

2^10|c^2=h,e^2=j,f^2=ij,c^a=ch,c^b=chi,e^b=ej,e^d=ehj,f^a=fij,f^c=fj,f^d=fg

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72232438, #72234767, #72234830, …
maximal normal subgroup of	#72232438, #72234767, #72234830, …
maximal quotient of	#72323731

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