#72332008

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 8987 conjugacy classes of subgroups; only the 3385 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)(5,6)(7,12)(8,10)(9,11)(21,22,25,26)(23,27,28,24);(5,7)(6,8)(9,10)(11,12)(13,14)(15,16)(17,20)(18,19)(22,26)(23,28);(2,4)(5,6)(7,12)(8,10)(9,11)(13,15)(14,18)(16,20)(17,19)(21,23,25,28)(22,27,26,24);(1,2,3,4)(13,15)(14,16)(17,19)(18,20)(21,24)(22,28)(23,26)(25,27);(5,6)(7,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,24)(22,23)(25,27)(26,28);(5,8)(6,10)(7,11)(9,12)(13,16)(14,19)(15,20)(17,18)(21,22)(23,27)(24,28)(25,26)

Polycyclic presentation:

2^10|a^2=i,c^2=i,d^2=ghi,b^a=bj,c^b=cg,d^a=dgh,d^c=dgh,f^a=fi,f^b=fg,f^d=fgj,f^e=fg

Conjugacy classes & character table

142 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72332061
maximal normal subgroup of	#72332061
maximal quotient of	#72333255

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