#135159

order	512
nilpotency class	3
isoclinism	isoclinic class of #8746
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#381
commutator	C₂⁴	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1817
prime radical	C₂⁵	C₂×C₈
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

664 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 664 conjugacy classes of subgroups — too many to draw. Only its 110 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) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=d,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,f^b=fi,e^c=ei,f^c=fg

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42355);