#72319366

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#45599
commutator	C₂²×C₄	C₂⁴×C₄
Frattini, cosocle	C₂³×C₄	C₂⁵
socle	C₂	#558844
prime radical	#7500918	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

4840 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:

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

Polycyclic presentation:

2^10|a^2=g,b^2=f,c^2=gij,d^2=gj,h^2=j,b^a=bghj,c^a=ci,c^b=cf,d^b=dfgh,d^c=dfgh,e^b=eghi,e^c=eghj,f^e=fj,g^b=gj,g^c=gj,h^e=hj,i^a=ij,i^b=ij,i^c=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319501
maximal normal subgroup of	#72319501

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