#502871

order	512
nilpotency class	4
isoclinism	isoclinic class of #501058
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#60693	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

880 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)(3,7)(4,5)(6,8)(9,10,14,17)(11,19,20,33)(12,24,25,39)(13,27,26,18)(15,28,30,22)(16,31,32,21)(23,38,36,35)(29,37,34,40);(2,5)(6,7)(11,20)(12,25)(16,32)(19,33)(21,31)(23,36)(24,39)(35,38);(9,11,21,37,13,23,25,28,14,20,31,40,26,36,12,22)(10,15,24,35,18,29,32,33,17,30,39,38,27,34,16,19);(1,3,4,8)(2,6,5,7)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410067);