#72287863

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	#7624085	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ #17330 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2183 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

103 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318539, #72318841, #72319144, …
maximal normal subgroup of	#72318539, #72318841, #72319144, …

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