#462662

order	512
nilpotency class	3
isoclinism	isoclinic class of #458126
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2420
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

827 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=e,d^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=di,d^c=de,e^c=ei,g^c=gi,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, 369858);