#133536

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, prime radical, cosocle	C₂²×#41	C₂³
LCS	G ▷ C₂²×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 678 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) = 128

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

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

Polycyclic presentation:

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

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, 40732);