#72336680

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 9972 conjugacy classes of subgroups; only the 3309 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:

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

Polycyclic presentation:

2^10|b^2=gh,e^2=gij,f^2=hi,b^a=bg,c^b=ci,d^b=dgij,e^a=ej,e^b=ehij,e^c=egh,e^d=egij,f^a=fgj,f^c=fghi,f^d=fghi,f^e=fg

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72336751
maximal normal subgroup of	#72336751

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