#72256708

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
prime radical	C₂×#52012	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

17007 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)(7,11)(9,12)(13,14)(15,18)(16,20)(17,19)(21,22,25,27)(23,24,26,28);(1,2,3,4)(5,6,8,10)(7,12,11,9)(13,15)(14,18)(16,19)(17,20)(21,23)(22,24)(25,26)(27,28);(5,6,8,10)(7,9,11,12)(13,15)(14,18)(16,19)(17,20);(5,7,8,11)(6,9,10,12)(13,16,17,18)(14,15,19,20);(5,7)(6,9)(8,11)(10,12)(13,16)(14,15)(17,18)(19,20)(21,24)(22,26)(23,27)(25,28);(5,6)(7,9)(8,10)(11,12)(13,14)(15,16)(17,19)(18,20)(21,24)(22,26)(23,27)(25,28)

Polycyclic presentation:

2^10|a^2=h,b^2=gi,c^2=i,d^2=j,b^a=bg,d^a=dj,d^b=di,e^a=ej,e^b=ehj,e^c=eij,e^d=ej,f^b=fhj,f^c=fj

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256713, #72256795, #72316598
maximal normal subgroup of	#72256713, #72256795, #72316598

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