#72334813

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	#3979362
commutator, Frattini, cosocle	C₂⁵	C₂⁵
prime radical	#401060	C₂
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

3399 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:

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

Polycyclic presentation:

2^10|b^2=f,c^2=i,d^2=h,b^a=bh,c^a=ch,d^a=dh,d^b=dg,d^c=df,e^a=eghi,e^c=ei,e^d=ei,f^a=fj,f^d=fj,g^a=gj,g^b=gj,g^d=gj,h^b=hj,h^c=hj,i^a=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72334815, #72334833, #72334842, …
maximal normal subgroup of	#72334815, #72334833, #72334842, …

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