#507620

order	512
nilpotency class	4
isoclinism	isoclinic class of #500591
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3253
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#29290
prime radical	C₂³	#575
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

418 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 158 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) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=eg,c^2=gh,d^2=e,e^2=i,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,h^a=hi,c^b=cg,f^c=fh,g^c=gi,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 414816);