#129311

order	512
nilpotency class	4
isoclinism	isoclinic class of #12727
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1782
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂	#8300
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #2873 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

641 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 641 conjugacy classes of subgroups — too many to draw. Only its 116 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36507);