#128716

order	512
nilpotency class	3
isoclinism	isoclinic class of #106004
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²

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	C₂³	#435
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₃₂ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

267 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 267 conjugacy classes of subgroups — too many to draw. Only its 122 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=eg,d^2=gi,e^2=h,f^2=g,h^2=i,b^a=bd,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, 35912);