#72258163

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#2433
commutator, Frattini, cosocle	C₂⁴×C₄	C₂⁴
prime radical	#6500028	C₂
LCS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

4981 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^10|a^2=h,b^2=i,c^2=fi,h^2=j,b^a=bhj,c^a=cj,c^b=cf,d^a=dg,d^b=de,d^c=de,e^c=ei,f^d=fi,h^b=hj

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

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