#72232983

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 9929 conjugacy classes of subgroups; only the 3317 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:

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

Polycyclic presentation:

2^10|c^2=g,e^2=h,f^2=gi,c^a=cg,c^b=ch,e^a=ehj,e^c=ei,e^d=eh,f^b=fgi,f^d=fg,f^e=fg

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72232993
maximal normal subgroup of	#72232993
maximal quotient of	#72331633

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