#72280697

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

At least 8781 conjugacy classes of subgroups; only the 3337 normal subgroups are shown.

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:

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

Polycyclic presentation:

2^10|a^2=h,b^2=gi,c^2=gj,d^2=gh,f^2=gh,b^a=bg,c^a=cgj,c^b=cg,d^a=dghi,d^b=dh,d^c=dhi,e^a=eg,e^b=egi,e^c=eg,e^d=eg,f^a=fg,f^d=fh,f^e=fg

Conjugacy classes & character table

142 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.7*F.9);;
gap> SetPower(coll, 3, F.7*F.10);;
gap> SetPower(coll, 4, F.7*F.8);;
gap> SetPower(coll, 6, F.7*F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
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*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.9);;
gap> SetConjugate(coll, 5, 3, F.5*F.7);;
gap> SetConjugate(coll, 5, 4, F.5*F.7);;
gap> SetConjugate(coll, 6, 1, F.6*F.7);;
gap> SetConjugate(coll, 6, 4, F.6*F.8);;
gap> SetConjugate(coll, 6, 5, F.6*F.7);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);