#72307348

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#1505
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂×#97	C₂⁴
prime radical	C₂⁴	#418
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#1726 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1409 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

(10,16,18,31)(12,26)(13,27)(15,29)(17,33,35,30)(19,38,32,20)(21,39,37,36)(22,23)(25,40)(28,34);(1,2,3,6,4,5,7,8)(9,10,13,19,15,20,12,16,14,18,26,32,29,38,27,31)(11,21,23,30,25,17,34,36,24,37,28,33,40,35,22,39);(2,5)(6,8)(9,11)(10,17,18,35)(12,22)(13,28)(14,24)(15,25)(16,30,31,33)(19,39,32,36)(20,37,38,21)(23,26)(27,34)(29,40);(2,5)(6,8)(10,18)(16,31)(17,35)(19,32)(20,38)(21,37)(30,33)(36,39)

Polycyclic presentation:

2^10|a^2=gj,b^2=f,c^2=gj,e^2=h,f^2=i,h^2=j,i^2=j,b^a=be,c^a=cgj,c^b=cij,d^b=dhi,e^a=eh,e^b=eg,f^a=fgh,f^c=fj,f^e=fj,g^b=gj,h^a=hj,i^a=ij

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307374, #72307544, #72307650
maximal normal subgroup of	#72307374, #72307544, #72307650

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