#72338543

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#56346
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
socle	C₂	#3866912
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

5886 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)(3,7)(4,14)(5,19,20,25)(6,8)(9,15)(10,21,13,22)(11,23)(12,17)(16,26,32,27)(18,24)(28,29,31,30);(2,8,3,11)(4,15,18,17)(10,19,13,25)(16,30,32,29);(1,3,7,2)(4,9)(5,19,20,25)(6,11,23,8)(10,21,13,22)(12,18)(14,17)(15,24)(16,31)(26,30)(27,29)(28,32);(1,4)(2,9)(3,12)(5,16)(6,15)(7,18)(8,24)(10,26)(11,14)(13,27)(17,23)(19,28)(20,32)(21,30)(22,29)(25,31);(1,5)(2,10)(3,13)(4,16)(6,22)(7,20)(8,25)(9,26)(11,19)(12,27)(14,28)(15,29)(17,30)(18,32)(21,23)(24,31)

Polycyclic presentation:

2^10|a^2=f,b^2=i,c^2=h,g^2=j,b^a=bg,c^a=cfj,c^b=cgj,d^a=dgj,d^b=dgj,d^c=dhij,e^a=eghij,e^b=ei,e^c=eghj,f^e=fj,g^a=gj,g^c=gj,g^d=gj,g^e=gj,h^d=hj,i^a=ij,i^c=ij,i^d=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338567
maximal normal subgroup of	#72338567

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