#72300566

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 9472 conjugacy classes of subgroups; only the 3453 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:

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

Polycyclic presentation:

2^10|a^2=h,b^2=g,c^2=g,b^a=bhi,c^a=cgj,c^b=cgj,d^a=dg,d^c=dg,e^a=eh,e^b=eg,e^c=eij,e^d=eg,f^a=fh,f^b=fgh,f^c=fg,f^d=fg

Conjugacy classes & character table

154 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72300575, #72329448
maximal normal subgroup of	#72300575, #72329448
maximal quotient of	#72331613

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