#492508

order	512
nilpotency class	4
isoclinism	isoclinic class of #492504
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

530 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|b^2=i,c^2=i,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fh,g^a=gi,d^b=de,e^b=eg,g^b=gi,f^c=fh,e^d=eg,g^d=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309422
maximal normal subgroup of	#72309422

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 399704);