#452671

order	512
nilpotency class	3
isoclinism	isoclinic class of #417541
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2487
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

655 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)(11,21,24,16)(12,25)(13,20)(14,18)(15,19)(17,27)(22,36,37,39)(23,32,29,31)(26,33)(28,34)(30,38,35,40);(1,3,4,7)(2,5,6,8)(9,11)(10,16)(12,22)(13,23)(14,29)(15,24)(17,30)(18,32)(19,21)(20,31)(25,39)(26,37)(27,38)(28,40)(33,36)(34,35);(3,7)(5,8)(9,12,13,27,15,26,14,28)(10,17,18,25,19,34,20,33)(11,22,23,38,24,37,29,40)(16,30,32,39,21,35,31,36);(10,18,19,20)(12,26)(16,31,21,32)(17,33,34,25)(22,37)(27,28)(30,39,35,36)(38,40)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 359867);