#146159

order	512
nilpotency class	4
isoclinism	isoclinic class of #143938
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#469
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#2087
prime radical	C₂³	#467
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

202 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 78 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 — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=d,c^2=dgh,e^2=gi,f^2=hi,g^2=i,b^a=bd,c^a=ce,f^a=fi,c^b=cf,f^b=fh,d^c=di,e^c=eg,f^c=fhi,g^c=gi

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, 53355);