#72285702

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	C₂×#1943
commutator	C₄×C₈	C₂³×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
prime radical	C₂⁴	#501
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂×#1451 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

769 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:

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

Polycyclic presentation:

2^10|a^2=g,b^2=e,c^2=eghi,e^2=h,f^2=hij,g^2=i,i^2=j,b^a=bgij,c^a=cij,c^b=cf,d^b=dh,e^a=ej,e^c=ej,f^b=fhij,f^c=fh,g^b=gi,g^c=gj,i^b=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286063
maximal normal subgroup of	#72286063

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