#72321313

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×#102
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₄×C₈	C₂⁵
prime radical	#10576902	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #3444 ◁ #60204 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

4819 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:

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

Polycyclic presentation:

2^10|a^2=f,d^2=h,f^2=h,g^2=i,i^2=j,c^a=cj,d^a=dj,d^b=dj,d^c=dhj,e^a=eg,e^c=ehj,e^d=ehj,f^e=fj,g^a=gi,g^e=gij,i^a=ij,i^e=ij

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321406, #72322669
maximal normal subgroup of	#72321406, #72322669

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