#72300304

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁰×C₄²

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

20127 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^10|a^2=h,b^2=g,c^2=gh,b^a=bi,c^a=cghij,c^b=cghij,d^c=di,e^a=eh,e^b=eg,e^c=egh,e^d=eg,f^b=fgh,f^c=fgh,f^d=fh,f^e=fgh

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72300310
maximal normal subgroup of	#72300310

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