#72279333

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#13348
commutator	C₂×#127	C₂²×C₄
Frattini, prime radical, cosocle	C₂²×#127	C₂³
LCS	G ▷ C₂×#127 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×C₄ ◁ C₂²×#127 ◁ G
LECS	G ▷ C₂²×#127 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

2385 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^10|a^2=j,b^2=g,c^2=ef,g^2=i,h^2=j,b^a=bg,c^a=cd,c^b=ce,d^b=dh,e^b=ehj,f^b=fhj,g^a=gi,g^c=gh,g^d=gi,g^e=gi,g^f=gi,h^a=hi,h^b=hij,h^c=hj,h^d=hj,h^e=hj,h^f=hj,i^c=ij

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72279422, #72279506, #72279660
maximal normal subgroup of	#72279422, #72279506, #72279660

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