#133683

order	512
nilpotency class	4
isoclinism	isoclinic class of #133390
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

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	#59001	C₂
LCS	G ▷ C₂²×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

981 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 981 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 24 points:

(1,2)(3,5)(4,7)(6,8)(9,10,13,18,14,17,12,16)(11,19,20,24,21,15,22,23);(2,5)(3,8)(6,7)(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,19)(18,24);(3,8)(9,12,14,13)(10,16,17,18)(11,20,21,22)(15,23,19,24)

Polycyclic presentation:

2^9|a^2=f,c^2=gi,d^2=i,f^2=gi,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, 40879);