#415205

order	512
nilpotency class	3
isoclinism	isoclinic class of #396724
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#397
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

750 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:

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

Polycyclic presentation:

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

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, 322401);