#497819

order	512
nilpotency class	3
isoclinism	isoclinic class of #188770
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#432
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #1770 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

437 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405015);