#133678

order	512
nilpotency class	4
isoclinism	isoclinic class of #133390
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1790
commutator	C₂²×D₄	C₂²×C₄
Frattini, cosocle	C₂×C₄×D₄	C₂³
prime radical	#2669	C₄
LCS	G ▷ C₂²×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

700 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 76 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) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 40874);