#72287293

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₆ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#442
commutator	C₈	C₂³×C₁₆
Frattini, cosocle	C₈²	C₂⁴
socle	C₂²	C₂×#1580
prime radical	C₂×#509	C₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ C₂×#1312 ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

715 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

(18,22)(23,28)(24,29)(30,31);(1,2,4,8,5,9,11,15,6,10,14,3,7,12,16,13)(18,23,22,28)(19,25)(20,27)(24,30,29,31)(26,32);(1,3,11,2,7,15,4,12,6,8,16,10,5,13,14,9)(18,24,22,29)(19,26)(23,30,28,31)(25,32);(17,18,19,23,20,24,25,31,21,22,26,28,27,29,32,30)

Polycyclic presentation:

2^10|b^2=e,c^2=eg,d^2=f,e^2=g,f^2=h,g^2=i,h^2=j,d^a=dj,d^b=df,d^c=dh,e^d=ej,f^b=fh,f^c=fj,h^b=hj

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320447, #72320725, #72321010, …
maximal normal subgroup of	#72320447, #72320725, #72321010, …
maximal quotient of	#72287304

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