#72287799

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	#10576894	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #3444 ◁ #60204 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

4782 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:

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

Polycyclic presentation:

2^10|b^2=j,d^2=f,f^2=h,g^2=i,i^2=j,b^a=bh,c^b=chj,d^a=dh,d^b=dj,d^c=dj,e^b=ej,e^d=eg,f^e=fj,g^d=gi,g^e=gij,i^d=ij,i^e=ij

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287809, #72287827, #72319058, …
maximal normal subgroup of	#72287809, #72287827, #72319058, …
maximal quotient of	#72343862

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