#417927

order	512
nilpotency class	3
isoclinism	isoclinic class of #417487
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1981 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)(3,7,8,5)(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);(1,3,4,8)(2,5,6,7)(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);(9,12)(10,18)(11,21)(13,26)(14,27)(15,29)(16,28)(17,32)(19,36)(20,37)(22,38)(23,39)(24,30)(25,31)(33,35)(34,40);(3,8)(5,7)(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,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, 325123);