#153502

order	512
nilpotency class	6
isoclinism	isoclinic class of #153496
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13569
commutator	#125	C₂²×C₄
Frattini, cosocle	#491	C₂³
prime radical	#3548	C₄
LCS	G ▷ #125 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #3139 ◁ G
LECS	G ▷ #491 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

299 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 299 conjugacy classes of subgroups — too many to draw. Only its 46 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:

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

Polycyclic presentation:

2^9|a^2=e,d^2=h,e^2=i,f^2=hi,g^2=i,b^a=bd,d^a=df,f^a=fg,g^a=gh,h^a=hi,c^b=chi,d^b=dh,e^b=efi,f^b=fhi,g^b=gh,d^c=di,e^d=egh,f^d=fh,g^d=gi,f^e=fhi,g^e=gi

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60698);