#72285387

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#46996
commutator, Frattini, cosocle	C₂⁵	C₂⁵
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

5709 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:

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

Polycyclic presentation:

2^10|a^2=f,c^2=fg,d^2=ij,b^a=bh,c^a=cg,c^b=cgi,d^a=dgij,d^b=dhi,d^c=dfj,e^a=efgj,e^b=ei,e^c=efg,f^b=fj,f^d=fj,g^d=gj,h^a=hj,h^c=hj,i^a=ij,i^c=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285704
maximal normal subgroup of	#72285704

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