#508379

order	512
nilpotency class	4
isoclinism	isoclinic class of #508286
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33582	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

703 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 703 conjugacy classes of subgroups — too many to draw. Only its 146 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)(9,10,15,17,14,20,12,18)(11,16,24,32,25,33,22,30)(13,19,28,34,27,37,26,35)(21,36,38,29,39,40,23,31);(2,5)(6,8)(9,11)(10,16)(12,22)(13,21)(14,25)(15,24)(17,32)(18,30)(19,36)(20,33)(23,26)(27,39)(28,38)(29,34)(31,35)(37,40);(3,7)(6,8)(10,17)(11,21)(12,15)(13,26)(16,29)(18,20)(22,38)(23,24)(25,39)(27,28)(30,40)(31,33)(32,36)(34,35);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

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

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, 415575);