#72337331

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁶
commutator	C₂³	C₂⁵×C₄
prime radical	C₂⁵×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

67033 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^10|a^2=i,c^2=j,f^2=g,b^a=bi,f^a=fi,f^b=fh,f^c=fhij,f^d=fhj,f^e=fij

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72337332
maximal normal subgroup of	#72337332
maximal quotient of	#72337357, #72337380, #72340045, …

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