#508536

order	512
nilpotency class	4
isoclinism	isoclinic class of #508492
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#481
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#469	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

518 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 132 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415732);