#418234

order	512
nilpotency class	3
isoclinism	isoclinic class of #417487
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
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

1277 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:

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

Polycyclic presentation:

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

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, 325430);