#408194

order	512
nilpotency class	3
isoclinism	isoclinic class of #400051
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1017 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)(3,8)(4,6)(5,7)(11,21)(12,26)(13,14)(16,22)(17,31)(18,19)(23,37)(24,25)(27,28)(29,39)(30,34)(32,33);(1,3,5,4)(2,6,7,8)(9,10)(11,22)(12,17)(13,18)(14,19)(15,20)(16,25)(21,30)(23,36)(24,34)(26,31)(27,32)(28,33)(29,38)(35,39)(37,40);(9,11)(10,16)(12,23)(13,25)(14,21)(15,24)(17,29)(18,34)(19,22)(20,30)(26,35)(27,38)(28,37)(31,36)(32,40)(33,39);(1,4,5,3)(2,6,7,8)(9,12,13,27,15,28,14,26)(10,17,18,32,20,33,19,31)(11,23,25,38,24,37,21,35)(16,29,34,40,30,39,22,36)

Polycyclic presentation:

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

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, 315390);