#145490

order	512
nilpotency class	4
isoclinism	isoclinic class of #145476
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2054
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#459
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

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

normal subgroups only

This group has 271 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) = 128

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

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

Polycyclic presentation:

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

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, 52686);