#511985

order	512
nilpotency class	4
isoclinism	isoclinic class of #511751
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

988 conjugacy classes of subgroups; only the 180 normal subgroups are shown.

normal subgroups only

This group has 988 conjugacy classes of subgroups — too many to draw. Only its 180 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) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ei,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,h^a=hi,f^c=fh,g^c=gi,h^c=hi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419181);