#396658

order	512
nilpotency class	3
isoclinism	isoclinic class of #394382
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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1053 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)(3,5)(4,6)(7,8)(9,10,14,19,15,20,16,31)(11,21,23,33,13,29,30,17)(12,24,26,39,27,18,28,35)(22,37,38,32,25,36,40,34);(1,3,4,7)(2,5,6,8)(9,11,15,13)(10,17,20,33)(12,25,27,22)(14,30,16,23)(18,34,24,32)(19,29,31,21)(26,38,28,40)(35,36,39,37);(2,6)(5,8)(9,12,14,26,15,27,16,28)(10,18,19,35,20,24,31,39)(11,22,23,38,13,25,30,40)(17,32,21,36,33,34,29,37);(9,13,15,11)(10,17,20,33)(12,25,27,22)(14,30,16,23)(18,32,24,34)(19,21,31,29)(26,40,28,38)(35,36,39,37)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=e,d^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,e^b=ei,g^c=gi,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, 303854);