#512059

order	512
nilpotency class	4
isoclinism	isoclinic class of #511753
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	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

973 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 973 conjugacy classes of subgroups — too many to draw. Only its 174 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)(3,6)(4,5)(7,8)(9,10,12,16)(11,19,20,38)(13,25,23,37)(14,28,24,18)(15,30,31,21)(17,34,33,27)(22,39,29,35)(26,36,40,32);(2,5)(6,8)(9,10)(11,15)(12,16)(13,17)(14,18)(19,21)(20,31)(22,32)(23,33)(24,28)(25,27)(26,35)(29,36)(30,38)(34,37)(39,40);(1,3,4,7)(2,6,5,8)(9,11,12,20)(10,15,16,31)(13,26,23,40)(14,29,24,22)(17,35,33,39)(18,36,28,32)(19,37,38,25)(21,34,30,27);(10,16)(13,23)(15,31)(18,28)(21,30)(26,40)(27,34)(32,36)

Polycyclic presentation:

2^9|a^2=i,c^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,h^a=hi,d^b=di,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, 419255);