#143389

order	512
nilpotency class	4
isoclinism	isoclinic class of #143217
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	C₂³	#447
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#446 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

374 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 84 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50585);