#72286787

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

4847 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)(19,30)(20,34)(21,27)(22,28)(23,25)(24,29)(26,35)(31,40)(32,42)(33,41)(36,48)(37,45)(38,46)(39,44)(43,47);(1,2)(3,8)(4,9)(5,10)(6,7)(11,15)(12,14)(13,16)(17,19,24,23)(18,25,29,30)(20,31,35,38)(21,32,22,33)(26,40,34,46)(27,41,28,42)(36,39,47,37)(43,44,48,45);(2,7)(8,14)(9,15)(10,16)(20,35)(26,34)(31,38)(36,47)(37,39)(40,46)(43,48)(44,45);(1,3,5,11,6,12,13,4)(2,8,10,15,7,14,16,9)(17,20,21,37,23,38,33,47,24,35,22,39,19,31,32,36)(18,26,27,44,25,40,41,48,29,34,28,45,30,46,42,43);(1,4,13,12,6,11,5,3)(2,9,16,14,7,15,10,8)(19,23)(20,36,35,47)(21,33)(22,32)(25,30)(26,43,34,48)(27,41)(28,42)(31,37,38,39)(40,45,46,44)

Polycyclic presentation:

2^10|b^2=j,d^2=f,e^2=fgh,f^2=h,g^2=i,h^2=j,i^2=j,b^a=bj,c^b=chij,d^a=dj,d^c=dj,e^b=ej,e^d=eg,f^e=fi,g^d=gj,g^e=gi,h^e=hj,i^e=ij

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286877, #72287030, #72318841, …
maximal normal subgroup of	#72286877, #72287030, #72318841, …

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