#135158

order	512
nilpotency class	3
isoclinism	isoclinic class of #8738
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	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1817
prime radical	C₂⁵	M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

700 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 114 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)(9,10,13,20,15,19,29,35,14,18,28,34,12,17,26,33)(11,21,24,36,25,32,38,40,22,16,27,31,23,30,37,39);(9,11)(10,16)(12,23)(13,27)(14,22)(15,25)(17,32)(18,21)(19,30)(20,36)(24,28)(26,38)(29,37)(31,34)(33,39)(35,40);(1,3)(2,5)(4,8)(6,7)(10,17,18,19)(11,22)(16,30,21,32)(20,33,34,35)(23,25)(24,27)(31,39,36,40)(37,38)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239364, #72239479, #72239480
maximal normal subgroup of	#72239364, #72239479, #72239480

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42354);