#411109

order	512
nilpotency class	3
isoclinism	isoclinic class of #393906
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2580
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

747 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)(3,8)(4,6)(5,7)(9,10,14,20,16,22,15,21)(11,13,24,31,25,32,19,17)(12,26,29,36,30,35,33,18)(23,37,27,28,39,34,38,40);(9,11,15,24,16,25,14,19)(10,17,21,13,22,31,20,32)(12,27,29,23,30,38,33,39)(18,28,26,37,36,40,35,34);(1,3,5,4)(2,6,7,8)(9,12,16,30)(10,18)(11,23)(13,28,32,40)(14,33,15,29)(17,34,31,37)(19,27)(20,35)(21,26)(22,36)(24,38)(25,39);(1,4,5,3)(2,6,7,8)(9,13,15,31,16,32,14,17)(10,19,21,11,22,24,20,25)(12,28,29,37,30,40,33,34)(18,27,26,23,36,38,35,39)

Polycyclic presentation:

2^9|a^2=e,b^2=gh,c^2=fg,d^2=f,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,e^c=ei,e^d=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, 318305);