#143556

order	512
nilpotency class	4
isoclinism	isoclinic class of #143510
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1999
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#3169	C₂²
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

381 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 381 conjugacy classes of subgroups — too many to draw. Only its 76 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=gi,e^2=hi,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,f^a=fhi,g^a=gi,c^b=cf,d^b=dg,e^b=ei,f^b=fhi,g^b=gi,d^c=dh,e^c=eh,f^c=fh

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, 50752);