#435116

order	512
nilpotency class	3
isoclinism	isoclinic class of #418769
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

799 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:

(1,2,5,4)(3,7,8,6)(9,10,14,19,15,20,31,38)(11,21,23,17,12,24,26,16)(13,18,30,37,29,36,28,35)(22,34,40,32,25,39,27,33);(1,3,5,8)(2,6,4,7)(9,11,15,12)(10,16,20,17)(13,27,29,40)(14,26,31,23)(18,34,36,39)(19,24,38,21)(22,30,25,28)(32,37,33,35);(1,4,5,2)(3,7,8,6)(9,12,15,11)(10,17,20,16)(13,25,29,22)(14,23,31,26)(18,33,36,32)(19,21,38,24)(27,30,40,28)(34,37,39,35);(3,8)(6,7)(9,13)(10,18)(11,22)(12,25)(14,30)(15,29)(16,32)(17,33)(19,37)(20,36)(21,39)(23,40)(24,34)(26,27)(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=de,e^b=ei,f^b=fi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 342312);