#72251689

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#2046052
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
prime radical	#417670	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

4095 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^10|b^2=f,c^2=fhj,d^2=ij,h^2=j,c^a=cfg,c^b=ci,d^a=dgj,d^b=dh,d^c=dgi,e^c=eghj,e^d=ei,f^c=fj,f^d=fj,g^c=gj,h^c=hj,h^e=hj,i^b=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72251715, #72251900, #72251931
maximal normal subgroup of	#72251715, #72251900, #72251931

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