#491773

order	512
nilpotency class	4
isoclinism	isoclinic class of #491760
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×D₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

488 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,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,d^c=di,f^c=fh,h^c=hi,e^d=eg

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398969);