#72283045

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³×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

31737 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:

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

Polycyclic presentation:

2^10|c^2=g,d^2=j,b^a=bi,c^b=cj,d^a=dj,d^c=dh,e^a=eg,e^b=eh,e^d=eg,f^b=fj,f^c=fg

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283161
maximal normal subgroup of	#72283161
maximal quotient of	#72292007

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