#72331734

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹×C₄

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

29741 conjugacy classes of subgroups.

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

Polycyclic presentation:

2^10|a^2=i,c^2=i,d^2=gj,b^a=bj,c^b=cg,d^a=dgij,d^c=dgij,f^b=fg,f^c=fi,f^d=fh,f^e=fg

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72331772
maximal normal subgroup of	#72331772
maximal quotient of	#72333218

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