#416910

order	512
nilpotency class	3
isoclinism	isoclinic class of #394464
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#394
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1064 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=fg,b^2=eh,d^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=di,e^c=ei,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 324106);