#72260738

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	#15948
commutator, Frattini, cosocle	C₂⁴×C₄	C₂⁴
prime radical	#6346698	C₂
LCS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

3659 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:

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

Polycyclic presentation:

2^10|a^2=e,b^2=gj,c^2=gj,d^2=gj,f^2=j,b^a=beg,c^a=cefj,d^a=dfh,d^b=dgh,d^c=dhj,e^b=ei,e^c=eij,e^d=ei,f^a=fi,f^b=fj,f^d=fij,g^a=gi,h^a=hj

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72260753
maximal normal subgroup of	#72260753

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