#72265570

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	C₂×C₄	#2673
commutator, Frattini, cosocle	C₂⁴×C₄	C₂⁴
socle	C₂²	#15368
prime radical	C₂²×#525	C₂²
LCS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

5975 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:

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

Polycyclic presentation:

2^10|a^2=e,b^2=g,d^2=f,e^2=j,b^a=bej,c^a=cgj,c^b=cgh,d^a=dgj,d^b=dehi,d^c=df,e^b=ej,f^a=fj,f^b=fj,g^d=gj,i^b=ij

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265580, #72265736, #72314310
maximal normal subgroup of	#72265580, #72265736, #72314310

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