#500098

order	512
nilpotency class	4
isoclinism	isoclinic class of #499916
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	#60444	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×M₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1048 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 1048 conjugacy classes of subgroups — too many to draw. Only its 160 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)(9,10)(11,20)(12,17)(13,25)(14,29)(15,19)(16,22)(18,28)(21,36)(23,37)(24,34)(26,39)(27,32)(30,35)(31,38)(33,40);(2,5)(6,8)(13,26)(16,31)(18,34)(22,38)(24,28)(25,39)(27,40)(32,33);(1,3,4,7)(2,6,5,8)(9,11)(10,16)(12,21)(13,27)(14,30)(15,23)(17,32)(18,35)(19,31)(20,25)(22,28)(24,38)(26,40)(29,33)(34,36)(37,39);(2,5)(6,8)(9,12,15,14)(10,17,19,29)(11,21,23,30)(13,24,26,28)(16,32,31,33)(18,25,34,39)(20,36,37,35)(22,27,38,40)

Polycyclic presentation:

2^9|c^2=e,d^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,h^a=hi,c^b=cg,f^b=fi,f^c=fh,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, 407294);