#72315217

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator	C₂×C₄	C₂³×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁵
socle	C₂²	C₂³×#97
prime radical	C₂×#532	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3315 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^10|c^2=g,e^2=f,g^2=i,i^2=j,b^a=bh,d^a=dj,d^b=dhj,d^c=dhj,e^c=eij,f^c=fj,g^e=gj

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72315218, #72315390, #72315568
maximal normal subgroup of	#72315218, #72315390, #72315568

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