#72286763

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#442
commutator	C₂×C₈	C₂⁴×C₄
Frattini, cosocle	C₄×C₈	C₂⁵
prime radical	C₂²×#509	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ #17323 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

2657 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:

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

Polycyclic presentation:

2^10|b^2=f,d^2=g,f^2=h,g^2=i,i^2=j,c^b=ch,d^a=dj,d^b=dg,d^c=dj,e^b=egij,e^d=eg,f^d=fj,f^e=fj,g^b=gi,g^e=gij,i^b=ij,i^e=ij

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286877, #72287030, #72287684, …
maximal normal subgroup of	#72286877, #72287030, #72287684, …

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