#72323402

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 10974 conjugacy classes of subgroups; only the 3637 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:

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

Polycyclic presentation:

2^10|a^2=h,c^2=g,d^2=i,f^2=g,b^a=bh,c^a=cg,c^b=cj,d^a=dhi,d^b=dh,e^a=ei,e^d=ei,f^b=fh,f^d=fh,f^e=fgh

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72323423, #72328270, C₂×#72323402
maximal normal subgroup of	#72323423, #72328270, C₂×#72323402
maximal quotient of	#72323745, #72331517, C₂×#72323402

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