#72319148

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator	C₂×C₄	C₂³×C₄²
Frattini, cosocle	C₄×C₈	C₂⁵
socle	C₂²	C₂³×#97
prime radical	C₂²×#41	M₁₆
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2558 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^10|b^2=g,c^2=f,d^2=h,f^2=h,g^2=i,i^2=j,b^a=bh,c^a=chj,c^b=chi,d^b=dj,d^c=dhj,e^b=ehj,e^d=eh,f^b=fj,g^c=gj

Conjugacy classes & character table

196 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319265, #72319316, #72319621, …
maximal normal subgroup of	#72319265, #72319316, #72319621, …

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