#72284602

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#2760
commutator, Frattini, cosocle	C₂⁵	C₂⁵
prime radical	C₂×#10336	C₂
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

4653 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^10|b^2=f,c^2=h,b^a=bh,c^a=ch,c^b=cg,d^a=dfgij,d^c=df,e^a=ej,e^c=ej,f^a=fj,f^c=fi,g^a=gj,g^b=gi,g^c=gj,h^b=hj,h^d=hi

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284709, #72338769, #72338905, …
maximal normal subgroup of	#72284709, #72338769, #72338905, …

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