#72341669

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂²	C₂⁸
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ G
LECS	G ▷ C₂² ▷ C₁

817673 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 8 generators on 36 points:

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

Polycyclic presentation:

2^10|d^2=j,b^a=bj,c^a=cj,d^a=dj,d^c=dj,e^a=ei,f^a=fi,f^d=fj,g^a=gij,g^e=gj,g^f=gj,h^c=hj

Conjugacy classes & character table

322 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341674, #72341809
maximal normal subgroup of	#72341674, #72341809

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