#72299669

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁵	C₂⁵
LCS	G ▷ C₂⁵ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

39107 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 24 points:

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

Polycyclic presentation:

2^10|a^2=h,b^2=h,c^2=fhi,d^2=fj,e^2=gh,b^a=bh,c^a=cf,c^b=cfi,d^a=dfhj,d^b=dfhj,d^c=dj,e^a=eg,e^b=egi,e^c=ehi,e^d=ehj

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72299673, #72336407, #72336416, …
maximal normal subgroup of	#72299673, #72336407, #72336416, …
maximal quotient of	#72299926

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