#72255648

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#16632
commutator, Frattini, cosocle	C₂⁴×C₄	C₂⁴
prime radical	C₂⁴×D₄	C₂³
LCS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

5983 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(5,6)(7,16)(8,21,23,13)(9,12)(10,14)(11,20)(15,25)(17,28)(18,31,33,36)(19,26)(22,29,24,30)(27,32,35,34);(1,2,4,3)(5,7,9,19)(6,11,12,28)(8,18,23,33)(10,17,25,20)(13,27,21,35)(14,16,15,26)(22,32,24,34)(29,36,30,31);(2,3)(6,12)(7,17)(8,22)(10,25)(11,26)(13,29)(16,28)(18,33)(19,20)(21,30)(23,24)(31,36);(1,3,4,2)(5,8,10,22,9,23,25,24)(6,13,15,29,12,21,14,30)(7,18,20,34,19,33,17,32)(11,27,16,31,28,35,26,36)

Polycyclic presentation:

2^10|a^2=h,b^2=e,d^2=eghj,f^2=j,g^2=j,b^a=bg,c^a=chi,c^b=cf,d^a=defghi,d^b=dhj,d^c=dfhij,f^b=fj,f^c=fj,g^a=gj,g^b=gj,g^c=gj,g^d=gj,h^d=hj,i^b=ij,i^d=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72255687, #72256056, #72256419
maximal normal subgroup of	#72255687, #72256056, #72256419

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