#72279989

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#18194
commutator	C₂×M₁₆	C₂³×C₄
Frattini, cosocle	C₂²×M₁₆	C₂⁴
prime radical	C₂×#33531	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

3224 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^10|a^2=e,b^2=g,c^2=h,e^2=i,f^2=j,i^2=j,b^a=bf,c^a=cehj,d^b=dhj,d^c=dj,e^b=eh,e^c=ei,f^a=fhj,f^b=fj,f^c=fhj,f^d=fj,f^e=fj,h^a=hj,i^c=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280113, #72313708, #72313737, …
maximal normal subgroup of	#72280113, #72313708, #72313737, …

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