#507730

order	512
nilpotency class	4
isoclinism	isoclinic class of #499952
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3205
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#29364
prime radical	#13340	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

529 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 529 conjugacy classes of subgroups — too many to draw. Only its 124 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)(3,13)(4,18)(5,11)(6,9)(7,12)(8,14)(10,15)(16,24)(17,23)(19,32)(20,25)(21,26)(22,29)(27,31)(28,30);(1,3,5,6,8,17,7,16)(2,9,12,13,14,24,11,23)(4,15,19,21,20,30,22,31)(10,25,27,32,28,18,26,29);(1,4)(2,10)(3,15)(5,19)(6,21)(7,22)(8,20)(9,18)(11,26)(12,27)(13,29)(14,28)(16,31)(17,30)(23,32)(24,25);(2,11,14,12)(9,23,24,13)(10,26,28,27)(18,32,25,29)

Polycyclic presentation:

2^9|b^2=e,d^2=g,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,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, 414926);