#496474

order	512
nilpotency class	4
isoclinism	isoclinic class of #33758
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×#102
prime radical	#60673	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

845 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)(9,10,13,19,14,20,28,34,15,17,29,35,27,36,12,18)(11,21,23,31,24,37,22,38,25,39,26,16,30,32,40,33);(1,2)(3,6)(4,5)(7,8);(2,5)(6,8)(9,11)(10,16)(12,26)(13,22)(14,30)(15,25)(17,31)(18,37)(19,21)(20,38)(23,28)(24,27)(29,40)(32,34)(33,36)(35,39);(1,3,4,7)(2,6,5,8)(10,17)(16,31)(18,34)(19,35)(20,36)(21,39)(32,37)(33,38)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403670);