#72287791

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, socle	C₂²	C₂²×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₄×C₈	C₂⁵
prime radical	#7623096	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ #17323 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2418 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^10|b^2=f,d^2=g,f^2=h,g^2=i,i^2=j,c^b=ch,d^a=dh,d^b=dg,d^c=dj,e^b=egij,e^d=eg,f^d=fj,f^e=fj,g^b=gi,g^e=gij,i^b=ij,i^e=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287809, #72287827, #72318707, …
maximal normal subgroup of	#72287809, #72287827, #72318707, …
maximal quotient of	#72343866

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