#145585

order	512
nilpotency class	4
isoclinism	isoclinic class of #144579
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#486
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#2041
prime radical	#3169	C₂²
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

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

normal subgroups only

This group has 303 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)(3,10)(4,12)(5,13)(6,9)(7,11)(8,15)(14,16)(17,18,23,29)(19,24,28,22)(20,25,21,30)(26,32,27,31);(1,3)(2,7)(4,8)(5,14)(6,12)(9,16)(10,13)(11,15)(17,19)(18,24)(20,26)(21,27)(22,29)(23,28)(25,32)(30,31);(1,4,5,7,6,10,15,16)(2,3,8,11,9,12,13,14)(18,25)(19,27)(20,31)(21,32)(22,24)(26,28)(29,30)

Polycyclic presentation:

2^9|a^2=i,c^2=d,d^2=g,e^2=hi,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fgh,h^a=hi,c^b=cf,d^b=dg,e^b=egh,f^b=fh,h^b=hi,e^c=egh,f^c=fgh,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, 52781);