#128718

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	#1752
prime radical	#532	C₂×C₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₃₂ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

313 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 313 conjugacy classes of subgroups — too many to draw. Only its 114 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)(9,10,14,18,15,21,31,37,13,20,28,36,29,19,38,35)(11,22,23,34,24,12,17,27,25,16,30,32,39,26,33,40);(1,2,4,8,5,7,3,6)(9,11,15,24,13,25,29,39)(10,16,19,12,20,22,21,26)(14,30,31,33,28,23,38,17)(18,34,35,40,36,32,37,27);(2,7)(6,8)(9,12,14,27,15,16,31,32,13,26,28,40,29,22,38,34)(10,17,18,25,21,30,37,39,20,33,36,11,19,23,35,24)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=e,d^2=gi,e^2=h,f^2=g,h^2=i,b^a=bd,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, 35914);