#72336732

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

32266 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:

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

Polycyclic presentation:

2^10|b^2=g,d^2=h,e^2=gh,f^2=ij,b^a=bj,c^a=cj,c^b=cgi,d^a=dj,d^b=dgh,d^c=dh,e^a=ej,e^b=eh,e^c=egh,e^d=eg,f^d=fgij,f^e=fg

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72336753
maximal normal subgroup of	#72336753

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