#72336729

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

At least 7395 conjugacy classes of subgroups; only the 3453 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:

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

Polycyclic presentation:

2^10|b^2=g,d^2=h,e^2=gh,f^2=ij,b^a=bj,c^b=cgi,d^a=dj,d^b=dgh,d^c=dg,e^a=ej,e^b=eh,e^c=eg,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, 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.7);;
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);;
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);