#415543

order	512
nilpotency class	3
isoclinism	isoclinic class of #397902
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#399
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

921 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 322739);