#417052

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

825 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)(9,10,14,18,15,19,13,28)(11,16,21,32,22,33,29,31)(12,23,27,35,25,39,26,17)(20,36,24,30,37,34,40,38);(2,5)(6,7)(9,11)(10,16)(12,24,25,40)(13,29)(14,21)(15,22)(17,34,35,36)(18,32)(19,33)(20,27,37,26)(23,38,39,30)(28,31);(9,12,13,26,15,25,14,27)(10,17,18,23,19,35,28,39)(11,20,21,24,22,37,29,40)(16,30,31,36,33,38,32,34);(1,3,4,8)(2,6,5,7)

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=f,d^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,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, 324248);