#72323391

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 6814 conjugacy classes of subgroups; only the 3713 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^10|c^2=g,d^2=i,e^2=hi,f^2=gh,c^a=cg,c^b=cj,d^b=dh,e^b=eh,e^d=eh,f^d=fi,f^e=fghi

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72323423, C₂×#72323391
maximal normal subgroup of	#72323423, C₂×#72323391
maximal quotient of	#72323732, C₂×#72323391

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