#146192

order	512
nilpotency class	4
isoclinism	isoclinic class of #143142
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#493
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#2093
prime radical	C₂²×D₈	Q₈
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

252 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=degi,d^2=gi,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,f^a=fghi,c^b=cf,d^b=dg,e^b=eg,g^b=gi,h^b=hi,d^c=dh,e^c=eh,g^c=gi,h^c=hi

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, 53388);