#426880

order	512
nilpotency class	3
isoclinism	isoclinic class of #418458
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

789 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=f,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,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, 334076);