#140380

order	512
nilpotency class	4
isoclinism	isoclinic class of #2212
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#459
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1947
prime radical	C₂³	#428
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

240 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 118 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=e,d^2=gi,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,d^b=dg,g^b=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47576);