#72280004

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

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

24601 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 24 points:

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

Polycyclic presentation:

2^10|a^2=g,b^2=h,c^2=i,d^2=h,f^2=g,c^a=chi,c^b=chi,d^a=dh,d^c=di,e^a=eghi,e^b=ehj,e^c=eghi,e^d=ehi,f^b=fh,f^e=fg

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280587, C₂×#72280004
maximal normal subgroup of	#72280587, C₂×#72280004
maximal quotient of	C₂×#72280004

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