#508560

order	512
nilpotency class	4
isoclinism	isoclinic class of #508494
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#481
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 413 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) = 128

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

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

Polycyclic presentation:

2^9|b^2=fh,c^2=e,d^2=ei,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,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, 415756);