#411251

order	512
nilpotency class	3
isoclinism	isoclinic class of #394746
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2398
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

670 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)(3,5)(4,6)(7,8)(9,10,13,16,15,20,14,19)(11,21,23,32,24,29,28,17)(12,18,25,30,27,34,26,33)(22,35,36,40,37,39,38,31);(1,3,4,7)(2,5,6,8)(10,16,20,19)(17,21,32,29)(18,30,34,33)(31,35,40,39);(9,11,14,23,15,24,13,28)(10,17,16,29,20,32,19,21)(12,22,26,36,27,37,25,38)(18,31,30,39,34,40,33,35);(2,6)(5,8)(9,12,14,26,15,27,13,25)(10,18,19,33,20,34,16,30)(11,22,23,36,24,37,28,38)(17,31,21,35,32,40,29,39)

Polycyclic presentation:

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

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, 318447);