#72328971

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 9224 conjugacy classes of subgroups; only the 3357 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:

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

Polycyclic presentation:

2^10|a^2=g,b^2=gh,d^2=ghi,f^2=hj,b^a=bgi,c^a=cj,c^b=cg,d^a=dhi,d^b=di,d^c=dj,e^a=eg,e^b=egh,e^c=ej,e^d=eghi,f^a=fi,f^b=fghij,f^c=fj,f^d=fi,f^e=fgi

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72329191
maximal normal subgroup of	#72329191

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