#72287715

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	#10580904	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×#41 ◁ C₂×C₈×#41 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

5554 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

142 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287735, #72287811, #72319190, …
maximal normal subgroup of	#72287735, #72287811, #72319190, …

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, 4, F.6);;
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*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.8*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.8*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);