#508447

order	512
nilpotency class	4
isoclinism	isoclinic class of #508313
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#472	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

532 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:

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

Polycyclic presentation:

2^9|a^2=ef,d^2=ei,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,g^b=gi,d^c=de,f^c=fh,g^c=gi,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415643);