#72285597

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×C₄×C₄⋊C₄
commutator	C₄²	C₂²×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂⁴
socle	C₂²	C₂×#1767
prime radical	C₂⁴	#432
LCS	G ▷ C₄² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#322 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1032 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:

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

Polycyclic presentation:

2^10|a^2=g,b^2=gh,d^2=e,e^2=h,f^2=hj,g^2=i,i^2=j,b^a=bf,c^a=chj,d^a=dij,d^b=dhi,e^a=ej,e^b=ej,f^a=fhj,f^b=fhj,g^d=gj

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285671, #72319682
maximal normal subgroup of	#72285671, #72319682

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