#72321907

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₈
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₄×C₈	C₂⁵
socle	C₂²	C₂²×C₄×D₈
prime radical	#7623228	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #2920 ◁ #17323 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2321 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)(3,6)(4,8)(5,7)(9,13)(10,12)(11,14)(15,16)(17,18)(19,22)(20,30)(21,36)(23,24)(25,33)(26,32)(27,37)(28,41)(29,38)(31,46)(34,43)(35,44)(39,48)(40,47)(42,45);(1,3,4,10,5,9,11,15)(2,6,8,12,7,13,14,16)(17,19,22,18,23,27,37,24)(20,31,34,39,35,28,42,25)(21,29,32,26,38,36,47,40)(30,33,45,41,44,48,43,46);(2,7)(3,9)(8,14)(10,15)(18,24)(19,27)(25,39)(26,40)(28,31)(29,36)(30,44)(43,45);(18,24)(19,27)(25,39)(26,40)(28,31)(29,36)(30,44)(43,45);(17,20,32,41,22,34,38,48,23,35,47,46,37,42,21,33)(18,25,36,45,27,31,40,44,24,39,29,43,19,28,26,30)

Polycyclic presentation:

2^10|b^2=g,e^2=fh,f^2=hj,g^2=i,h^2=j,b^a=bhj,c^a=ci,c^b=ci,d^a=dj,d^b=dj,e^a=ef,e^b=ej,f^a=fh,g^a=gj,h^a=hj

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321958, #72322516, #72322555
maximal normal subgroup of	#72321958, #72322516, #72322555

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