#129591

order	512
nilpotency class	4
isoclinism	isoclinic class of #12767
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1790
commutator	C₂×D₄	C₂²×C₈
Frattini, cosocle	C₂×#131	C₂³
socle	C₂	#8316
prime radical	#3481	C₄
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

619 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 619 conjugacy classes of subgroups — too many to draw. Only its 110 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:

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

Polycyclic presentation:

2^9|a^2=f,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,e^b=egi,f^b=fg,d^c=dg,f^c=fi,e^d=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, 36787);