#128580

order	512
nilpotency class	3
isoclinism	isoclinic class of #105688
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₄×C₂²⋊C₄
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1749
prime radical	C₂²×D₄	M₁₆
LCS	G ▷ C₄² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

406 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 406 conjugacy classes of subgroups — too many to draw. Only its 128 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35776);