#146245

order	512
nilpotency class	4
isoclinism	isoclinic class of #144123
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#493
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
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

299 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 299 conjugacy classes of subgroups — too many to draw. Only its 116 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) = 512

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

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

Polycyclic presentation:

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

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, 53441);