#146023

order	512
nilpotency class	4
isoclinism	isoclinic class of #143938
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#469
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#2076
prime radical	C₂²×D₄	Q₁₆
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

248 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 74 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:

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

Polycyclic presentation:

2^9|a^2=e,c^2=dg,e^2=gi,f^2=h,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=fh,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, 53219);