#128582

order	512
nilpotency class	3
isoclinism	isoclinic class of #106004
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 ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 404 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)(11,15,20,27)(12,22)(13,16)(14,18)(17,23)(19,29,32,26)(21,35,34,28)(24,37)(25,30)(31,38)(33,39,36,40);(1,2,4,7,5,8,3,6)(9,11,12,21,14,19,24,33,13,20,23,34,25,32,38,36)(10,15,17,28,18,29,31,40,16,27,22,35,30,26,37,39);(10,16)(11,19,20,32)(12,23)(15,26,27,29)(18,30)(21,36,34,33)(24,38)(28,40,35,39)

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=fg,c^b=cgh,d^b=di,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, 35778);