#138212

order	512
nilpotency class	4
isoclinism	isoclinic class of #138198
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1874
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂⁴	#102
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

433 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 433 conjugacy classes of subgroups — too many to draw. Only its 100 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfghi,d^2=ghi,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,c^b=cg,d^b=dh,e^b=ei,f^b=fh,h^b=hi,d^c=di,f^c=fi

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, 45408);