#492515

order	512
nilpotency class	4
isoclinism	isoclinic class of #492512
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

518 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)(9,10,14,19)(11,20,21,32)(12,25,28,34)(13,16,27,18)(15,30,31,23)(17,35,37,26)(22,38,39,33)(24,40,29,36);(9,11,14,21)(10,15,19,31)(12,22,28,39)(13,29,27,24)(16,33,18,38)(17,36,37,40)(20,34,32,25)(23,35,30,26);(2,5)(3,8)(6,7)(11,21)(15,31)(20,32)(22,39)(23,30)(24,29)(33,38)(36,40);(10,16)(11,22)(12,26)(13,27)(15,32)(18,19)(20,31)(21,39)(23,24)(25,34)(28,35)(29,30)(33,36)(38,40)

Polycyclic presentation:

2^9|a^2=i,b^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,c^b=ci,d^b=de,e^b=eg,g^b=gi,f^c=fh,e^d=eg,g^d=gi

Conjugacy classes & character table

65 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, 399711);