#72340220

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#2760
commutator, Frattini, cosocle	C₂⁵	C₂⁵
prime radical	#233549	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

3269 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,9)(4,7,13,15)(5,8)(6,11)(10,14,12,16)(17,18)(19,27,30,24)(20,31,34,25)(21,35,37,26)(22,23)(28,40)(29,39)(32,38)(33,43)(36,42)(41,46)(44,48,45,47);(18,23)(24,27)(25,31)(26,35)(38,40)(39,42)(41,43)(47,48);(1,3)(2,6,8,9)(4,12)(5,11)(7,14,15,16)(10,13)(17,19)(18,24)(20,32)(21,36,37,29)(22,30)(23,27)(25,40)(26,42,35,39)(28,34)(31,38)(33,45,46,44)(41,48,43,47);(1,4)(2,7,8,15)(3,10)(5,13)(6,14,9,16)(11,12)(17,20)(18,25)(19,28)(21,33,37,46)(22,34)(23,31)(24,38)(26,43,35,41)(27,40)(29,44,36,45)(30,32)(39,48,42,47);(6,9)(10,12)(17,21)(18,26)(19,29)(20,33)(22,37)(23,35)(24,39)(25,41)(27,42)(28,44)(30,36)(31,43)(32,45)(34,46)(38,47)(40,48)

Polycyclic presentation:

2^10|a^2=f,c^2=g,d^2=g,b^a=bj,c^a=ch,d^a=dfgh,d^c=dgi,e^a=eh,e^c=efg,e^d=ehj,f^c=fj,f^d=fi,f^e=fj,g^a=gij,g^e=gj,h^a=hj,h^c=hij,h^d=hj

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340246, #72340388, #72340391, …
maximal normal subgroup of	#72340246, #72340388, #72340391, …

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