#499630

order	512
nilpotency class	4
isoclinism	isoclinic class of #499263
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂³	C₂×AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

449 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 449 conjugacy classes of subgroups — too many to draw. Only its 136 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 — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=f,d^2=e,g^2=hi,h^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,g^b=gh,h^b=hi,d^c=dh,e^c=ei,g^c=gh,h^c=hi,g^d=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406826);