#145221

order	512
nilpotency class	4
isoclinism	isoclinic class of #145204
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2033
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#480
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 279 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=d,d^2=g,e^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=cf,d^b=dg,e^c=egh,f^c=fg,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, 52417);