#72331200

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁶
commutator	C₂³	C₂⁵×C₄
prime radical	C₂×#35346	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

At least 9929 conjugacy classes of subgroups; only the 6073 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:

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

Polycyclic presentation:

2^10|d^2=gh,b^a=bj,c^b=cg,d^a=di,d^c=di,f^a=fi,f^b=fg,f^d=fg,f^e=fg

Conjugacy classes & character table

202 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72331261, #72331723
maximal normal subgroup of	#72331261, #72331723
maximal quotient of	#72333246

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