#453028

order	512
nilpotency class	3
isoclinism	isoclinic class of #417570
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2482
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

776 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)(9,10,14,18)(11,20)(12,25)(13,16,29,19)(15,24)(17,26)(21,36,38,40)(22,32)(23,30)(27,35)(28,33)(31,39,34,37);(1,3,4,7)(2,5,6,8)(9,11)(10,15)(12,21)(13,22)(14,24)(16,30)(17,34)(18,20)(19,32)(23,29)(25,40)(26,37)(27,39)(28,38)(31,35)(33,36);(3,7)(5,8)(10,16,18,19)(15,30,20,32)(17,33,35,25)(31,40,34,36);(9,12,13,26,14,28,29,27)(10,17,16,33,18,35,19,25)(11,21,22,37,24,38,23,39)(15,31,30,40,20,34,32,36)

Polycyclic presentation:

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

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, 360224);