#72236944

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#14717
commutator, Frattini, cosocle	C₂⁴×C₄	C₂⁴
prime radical	C₂×#3484	C₂²
LCS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

4301 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^10|a^2=g,b^2=gij,e^2=j,b^a=bgh,c^a=ce,c^b=cef,d^a=dh,d^b=dhi,e^a=ei,e^b=ei,e^c=ej,e^d=ej,f^a=fi,g^c=gij,h^c=hj

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72236950, #72236971, #72237012
maximal normal subgroup of	#72236950, #72236971, #72237012

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