#142158

order	512
nilpotency class	4
isoclinism	isoclinic class of #125076
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#102
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

438 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 114 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)(3,7)(4,5)(6,8)(9,10,12,15,13,16,11,14)(17,18,23,27,20,28,30,25,22,26,31,32,24,19,21,29);(2,5)(6,7)(10,14,16,15)(17,19,20,26,22,28,24,18)(21,29,23,32,30,25,31,27);(1,3,4,8)(2,6,5,7)(17,20,22,24)(18,19,26,28)(21,23,30,31)(25,27,29,32)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 49354);