#394476

order	512
nilpotency class	3
isoclinism	isoclinic class of #394113
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1918 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)(12,26)(13,19)(14,18)(15,20)(16,24)(17,27)(22,25)(23,31)(28,33)(29,32)(30,34)(35,38)(36,37)(39,40);(1,3,4,8)(2,5,6,7)(9,10)(11,22)(12,17)(13,18)(14,19)(15,20)(16,24)(21,25)(23,36)(26,27)(28,34)(29,32)(30,33)(31,37)(35,39)(38,40);(9,11)(10,16)(12,23)(13,29)(14,24)(15,25)(17,31)(18,22)(19,21)(20,32)(26,36)(27,38)(28,39)(30,37)(33,35)(34,40);(9,12,15,28)(10,17,20,34)(11,23,25,39)(13,27,14,30)(16,31,32,40)(18,26,19,33)(21,35,22,36)(24,37,29,38)

Polycyclic presentation:

2^9|a^2=e,b^2=e,d^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,g^b=gi,f^c=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, 301672);