#491821

order	512
nilpotency class	4
isoclinism	isoclinic class of #491728
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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×SD₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

459 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309493
maximal normal subgroup of	#72309493

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 399017);