#135267

order	512
nilpotency class	3
isoclinism	isoclinic class of #8760
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	C₂×C₄×C₈	C₂³
socle	C₂²	#1817
prime radical	C₂⁴	C₂²⋊C₈
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 396 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)(3,5)(4,6)(7,8)(9,10,13,20,14,17,29,33,15,18,28,34,12,19,26,35)(11,21,24,36,25,30,38,39,22,16,27,31,23,32,37,40);(2,5)(6,7)(9,11)(10,16)(12,23)(13,27)(14,25)(15,22)(17,32)(18,21)(19,30)(20,36)(24,28)(26,38)(29,37)(31,34)(33,39)(35,40);(2,6)(4,8)(5,7)(10,17,18,19)(11,22)(13,28)(16,30,21,32)(20,35,34,33)(23,25)(26,29)(31,40,36,39)

Polycyclic presentation:

2^9|a^2=f,c^2=dg,e^2=gi,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=eg,f^c=fi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42463);