#72297901

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

21828 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 28 points:

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

Polycyclic presentation:

2^10|b^2=g,d^2=gh,e^2=g,f^2=gij,b^a=bj,c^b=ch,d^a=dghi,d^b=dh,d^c=dh,e^a=eg,e^c=ei,e^d=eh,f^b=fhij,f^d=fhij,f^e=fhij

Conjugacy classes & character table

166 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72297910
maximal normal subgroup of	#72297910
maximal quotient of	#72298609

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