#72319656

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#21503
commutator	C₄×C₈	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
prime radical	#490756	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×#97 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

888 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^10|a^2=e,c^2=ei,d^2=gi,e^2=i,f^2=h,g^2=i,h^2=j,b^a=bf,c^b=chj,d^a=dg,d^b=di,e^b=ej,f^a=fh,f^b=fhj,f^c=fj,g^a=gi,h^a=hj,h^b=hj

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319684, #72320294, #72320730, …
maximal normal subgroup of	#72319684, #72320294, #72320730, …

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