#474321

order	512
nilpotency class	3
isoclinism	isoclinic class of #459487
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	H(3,𝔽₄)
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2804
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

695 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:

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

Polycyclic presentation:

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

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, 381517);