#131122

order	512
nilpotency class	4
isoclinism	isoclinic class of #130734
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1786
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#446 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

550 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 550 conjugacy classes of subgroups — too many to draw. Only its 98 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=g,d^2=hi,f^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=ei,f^b=fgh,h^b=hi,d^c=di,f^c=fg

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 38318);