#398062

order	512
nilpotency class	3
isoclinism	isoclinic class of #394358
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1331 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:

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

Polycyclic presentation:

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

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, 305258);