#72224751

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×#127
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
socle	C₂²	C₂²×#392
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

37763 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^10|d^2=hj,h^2=j,b^a=bf,d^a=dh,d^b=dg,d^c=dj,e^b=ei,e^c=ej,f^d=fj,g^a=gj,h^a=hj

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257344
maximal normal subgroup of	#72257344
maximal quotient of	#72264125

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