#72238941

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#7555
commutator	C₂³×C₄	C₂×C₄²
Frattini, cosocle	C₂⁵×C₄	C₂³
prime radical	C₂³×#121	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵×C₄ ◁ G
LECS	G ▷ C₂⁵×C₄ ▷ C₂² ▷ C₁

5231 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^10|a^2=d,b^2=e,c^2=ehi,e^2=j,f^2=j,h^2=j,b^a=bg,c^a=cf,c^b=ch,d^b=dj,d^c=di,e^c=ei,f^a=fij,f^b=fj,g^a=gj,h^a=hj,h^b=hij,h^c=hi

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238951
maximal normal subgroup of	#72238951
maximal quotient of	#72238957

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