#129259

order	512
nilpotency class	4
isoclinism	isoclinic class of #12707
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂²×C₄⋊C₄	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

867 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 867 conjugacy classes of subgroups — too many to draw. Only its 134 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) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2,3,5,4,6,7,8)(9,10)(11,19,22,34)(12,16)(13,25,27,17)(14,18)(15,21)(20,35,37,33)(23,32,26,30)(24,31)(28,36)(29,39)(38,40);(9,11)(10,15,18,29)(12,20)(13,21)(14,22)(16,28,31,40)(17,19,25,34)(23,36)(24,37)(26,38)(27,39)(30,35,32,33);(9,12,14,24)(10,16,18,31)(11,20,22,37)(13,26,27,23)(15,28,29,40)(17,32,25,30)(19,33,34,35)(21,38,39,36)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36455);