#72251695

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#3089911
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
prime radical	#394249	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3419 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)(3,12,15,26)(4,18)(5,10)(6,11)(7,17,23,16)(8,21)(9,13)(14,27,28,32)(19,24)(20,25)(22,30,31,29);(1,3,13,16,6,15,5,17)(2,7,10,12,11,23,9,26)(4,14,19,29,21,28,20,30)(8,22,24,27,18,31,25,32);(1,4,13,19,6,21,5,20)(2,8,9,25,11,18,10,24)(3,14,17,30,15,28,16,29)(7,22,12,27,23,31,26,32);(8,18)(14,28)(24,25)(29,30);(3,15)(4,19)(5,13)(8,24)(9,10)(12,26)(14,29)(18,25)(20,21)(22,32)(27,31)(28,30)

Polycyclic presentation:

2^10|a^2=f,b^2=fgj,c^2=fgi,g^2=j,h^2=j,b^a=bh,c^a=cfj,c^b=cghij,d^a=dghi,d^b=dghi,d^c=dij,e^a=eghi,e^b=eghj,e^c=eg,f^b=fj,g^a=gj,g^b=gj,g^e=gj,h^c=hj,h^e=hj,i^a=ij,i^b=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72251715, #72251931, #72251940
maximal normal subgroup of	#72251715, #72251931, #72251940

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