#72338529

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#704054
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
prime radical	C₂×#23872	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

5844 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:

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

Polycyclic presentation:

2^10|b^2=g,c^2=gij,g^2=j,b^a=bh,c^a=ci,c^b=cij,d^a=df,d^b=dhj,d^c=dg,e^b=eij,e^c=ei,e^d=egh,f^e=fj,g^a=gj,g^c=gj,g^d=gj,g^e=gj,h^b=hj,h^c=hj,i^b=ij,i^c=ij,i^d=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338530
maximal normal subgroup of	#72338530

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