#415740

order	512
nilpotency class	3
isoclinism	isoclinic class of #396611
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

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

717 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:

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

Polycyclic presentation:

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

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, 322936);