#145608

order	512
nilpotency class	4
isoclinism	isoclinic class of #145594
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2041
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#486
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

278 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 72 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) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52804);