#415480

order	512
nilpotency class	3
isoclinism	isoclinic class of #394465
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×#397
prime radical	C₂²×#41	C₂³
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)(9,10,14,19,15,20,13,18)(11,16,22,31,23,32,29,36)(12,24,28,17,26,35,27,34)(21,33,25,39,37,40,38,30);(9,11)(10,16)(12,25,26,38)(13,29)(14,22)(15,23)(17,33,34,40)(18,36)(19,31)(20,32)(21,28,37,27)(24,30,35,39);(9,12)(10,17)(11,21)(13,28)(14,27)(15,26)(16,30)(18,35)(19,24)(20,34)(22,38)(23,37)(25,29)(31,40)(32,39)(33,36);(1,3,4,8)(2,5,6,7)

Polycyclic presentation:

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

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, 322676);