#421406

order	512
nilpotency class	3
isoclinism	isoclinic class of #418518
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

837 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

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, 328602);