#72234103

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#6628896
commutator, Frattini, cosocle	C₂×#41	C₂⁵
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #3498 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

At least 6238 conjugacy classes of subgroups; only the 947 normal subgroups are shown.

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:

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

Polycyclic presentation:

2^10|a^2=ij,d^2=j,h^2=j,c^b=cj,d^a=dh,d^b=dg,d^c=df,e^a=eij,e^c=ej,f^a=fi,f^b=fij,g^c=gi,g^f=gj,h^c=hij,h^d=hj,h^e=hj,h^f=hj,i^d=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72260462, #72260582
maximal normal subgroup of	#72260582
maximal quotient of	#72234104, #72260982, #72261713

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