#414674

order	512
nilpotency class	3
isoclinism	isoclinic class of #393979
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

1537 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)(11,21)(12,25)(13,19)(14,18)(15,20)(16,23)(17,27)(22,30)(24,35)(26,34)(28,33)(29,31)(32,38)(36,37)(39,40);(9,11)(10,16)(12,22)(13,29)(14,23)(15,24)(17,32)(18,35)(19,21)(20,31)(25,36)(26,39)(27,37)(28,38)(30,34)(33,40);(9,12,15,28)(10,17,20,34)(11,22,24,38)(13,27,14,26)(16,30,31,32)(18,25,19,33)(21,36,35,40)(23,39,29,37);(1,3,4,8)(2,5,6,7)

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=h,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,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, 321870);