#72249751

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#7557
commutator	C₂×C₄²	C₂×C₄²
Frattini, cosocle	C₂³×C₄²	C₂³
prime radical	C₂⁵	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄² ▷ C₂² ▷ C₁

2441 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^10|a^2=h,b^2=e,c^2=f,d^2=ij,g^2=j,h^2=j,b^a=bg,c^a=cf,c^b=cd,d^a=dj,d^b=di,e^c=ej,f^b=fij,g^b=gj,g^c=gi,h^b=hj

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249760, #72249794, #72249896
maximal normal subgroup of	#72249760, #72249794, #72249896

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