#414751

order	512
nilpotency class	3
isoclinism	isoclinic class of #394271
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1388 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)(11,15,21,30)(12,22,24,33)(13,26,28,18)(16,27,32,25)(19,34,23,39)(20,36,38,31)(29,37,40,35);(9,11)(10,15)(12,23,24,19)(13,20)(14,21)(16,29,32,40)(17,30)(18,31)(22,34,33,39)(25,37,27,35)(26,36)(28,38);(9,12)(10,16)(11,19)(13,27)(14,24)(15,29)(17,32)(18,33)(20,37)(21,23)(22,26)(25,28)(30,40)(31,39)(34,36)(35,38);(1,3,4,8)(2,5,6,7)(10,17)(15,30)(16,32)(18,26)(22,33)(29,40)(31,36)(34,39)

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=hi,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,g^b=gi,e^c=ei,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, 321947);