#72297891

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 6052 conjugacy classes of subgroups; only the 3349 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)(7,10)(8,9)(11,12)(13,14)(15,16)(17,19)(18,20)(22,24)(26,27);(2,4)(13,14)(15,20)(16,18)(17,19)(21,22)(23,27)(24,25)(26,28);(7,11)(10,12)(13,15)(14,16)(17,18)(19,20);(1,2,3,4)(6,8)(10,12)(15,18)(16,20)(21,23,25,28)(22,26,24,27);(2,4)(5,7)(6,10)(8,12)(9,11)(13,14)(15,20)(16,18)(17,19)(21,23)(22,27)(24,26)(25,28);(1,2)(3,4)(5,8,9,6)(7,12,11,10)(13,16,17,20)(14,18,19,15)(21,24,25,22)(23,27,28,26)

Polycyclic presentation:

2^10|d^2=ghj,f^2=gi,b^a=bg,c^b=ch,d^a=dhi,d^b=dghj,d^c=dh,e^c=ei,e^d=eghj,f^a=fgi,f^b=fghj,f^d=fhij,f^e=fghj

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72297910, #72323473, #72336887
maximal normal subgroup of	#72297910, #72323473, #72336887
maximal quotient of	#72298599, #72323765

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