#72234008

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 10587 conjugacy classes of subgroups; only the 3321 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:

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

Polycyclic presentation:

2^10|a^2=g,f^2=i,c^b=cgh,d^a=dg,d^c=dj,e^a=eh,e^b=ehi,f^c=fj,f^d=fi

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234056
maximal normal subgroup of	#72234056

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