#478913

order	512
nilpotency class	3
isoclinism	isoclinic class of #476095
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	H(3,𝔽₄)
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2684
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

819 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=ef,c^2=ef,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=def,e^b=ei,f^b=fi,h^b=hi,d^c=de,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 386109);