#135204

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

520 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 98 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)(9,10,13,19,14,20,23,16,15,18,29,32,30,37,11,21)(12,24,27,39,28,40,25,34,26,17,33,35,38,36,22,31);(9,11,14,13,15,23,30,29)(10,16,20,32,18,21,37,19)(12,25,28,33,26,22,38,27)(17,34,36,35,24,31,40,39);(1,3)(2,5)(4,8)(6,7)(9,12)(10,17,18,24)(11,22)(13,27)(14,28)(15,26)(16,31,21,34)(19,35,32,39)(20,36,37,40)(23,25)(29,33)(30,38)

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=d,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,c^b=ci,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, 42400);