#128569

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

420 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 124 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)(4,7)(6,8)(9,10)(11,15,20,28)(12,23)(13,16)(14,18)(17,24)(19,32,22,27)(21,36,35,29)(25,38)(26,30)(31,33)(34,39,37,40);(2,5,7,6)(9,11,12,21,14,22,25,37,13,20,24,35,26,19,33,34)(10,15,17,29,18,27,31,39,16,28,23,36,30,32,38,40);(10,16)(11,19,20,22)(12,24)(15,27,28,32)(18,30)(21,37,35,34)(25,33)(29,40,36,39)

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=ch,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, 35765);