#491444

order	512
nilpotency class	4
isoclinism	isoclinic class of #491439
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#493
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#33717	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

615 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)(3,7)(6,8)(10,15)(12,16)(13,23)(17,25)(18,26)(19,33)(21,29)(22,35)(24,30)(27,28)(31,40)(32,38)(34,39)(36,37);(9,10,16,28,13,19,24,26,14,18,30,33,23,27,12,15)(11,17,29,37,22,31,34,38,20,32,39,40,35,36,21,25);(1,2,3,5,4,6,7,8)(9,11)(10,17)(12,21)(13,22)(14,20)(15,25)(16,29)(18,32)(19,31)(23,35)(24,34)(26,38)(27,36)(28,37)(30,39)(33,40);(10,18)(11,20)(15,26)(19,27)(21,34)(22,35)(28,33)(29,39)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398640);