#138684

order	512
nilpotency class	4
isoclinism	isoclinic class of #12483
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#459
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1879
prime radical	C₂×#135	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

327 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 327 conjugacy classes of subgroups — too many to draw. Only its 90 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) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 40 points:

(2,5)(3,7)(6,8)(9,10,12,20,15,22,29,39,14,21,28,18,31,36,13,19)(11,23,24,38,27,37,25,16,26,34,30,17,32,33,40,35);(1,2,3,5,4,6,7,8)(9,11,14,26)(10,16)(12,25,28,40)(13,30,29,24)(15,32,31,27)(17,36)(18,37)(19,23)(20,33)(21,35)(22,38)(34,39);(2,6)(5,8)(9,11)(10,17)(12,25)(13,30)(14,26)(15,32)(16,22)(18,34)(19,37)(20,23)(21,38)(24,29)(27,31)(28,40)(33,39)(35,36)

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,d^2=ghi,e^2=hi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,c^b=cg,d^b=dh,e^b=eh,f^b=fh,h^b=hi,d^c=dh,e^c=eh,f^c=fh,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45880);