#72249841

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#24033
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	#6465638	C₂
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #20438 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2483 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^10|a^2=g,b^2=g,c^2=ghj,d^2=eg,f^2=i,h^2=ij,i^2=j,c^a=cf,c^b=cfhi,d^a=de,d^b=dg,d^c=dj,e^c=ej,f^a=fi,f^b=fj,f^c=fj,f^d=fj,g^c=gj,h^a=hij,h^b=hj,h^c=hj,i^a=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249855, #72252652, #72252755
maximal normal subgroup of	#72249855, #72252652, #72252755

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