#146233

order	512
nilpotency class	4
isoclinism	isoclinic class of #144123
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#493
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

381 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 381 conjugacy classes of subgroups — too many to draw. Only its 124 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=deg,d^2=gi,e^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ce,c^b=cf,d^b=dg,e^b=eg,f^b=fh,g^b=gi,d^c=dg,e^c=eg,f^c=fh,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53429);