#129777

order	512
nilpotency class	4
isoclinism	isoclinic class of #129777
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#8328
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	#60461	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#568 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

886 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 886 conjugacy classes of subgroups — too many to draw. Only its 106 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) = 32

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

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

Polycyclic presentation:

2^9|a^2=f,d^2=hi,e^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,h^a=hi,d^b=dh,f^b=fghi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,f^d=fi

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, 36973);