#72327042

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

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

24591 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^10|b^2=h,f^2=ghj,b^a=bi,c^b=cg,d^b=dghij,e^a=eghij,e^b=eh,e^c=ej,e^d=eghij,f^a=fhi,f^b=fgh,f^c=fj,f^d=fij,f^e=fj

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72327062, #72329811
maximal normal subgroup of	#72327062, #72329811

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