#152251

order	512
nilpotency class	5
isoclinism	isoclinic class of #149794
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2126
commutator	C₂×C₄²	C₂²×C₄
Frattini, prime radical, cosocle	#532	C₂³
socle	C₂	#12798
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #2931 ◁ G
LECS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

305 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 305 conjugacy classes of subgroups — too many to draw. Only its 68 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=d,c^2=h,d^2=i,e^2=h,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gh,h^a=hi,e^c=ehi,f^c=fghi,g^c=gi,f^e=fhi,g^f=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59447);