#135638

order	512
nilpotency class	3
isoclinism	isoclinic class of #8790
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#382
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1831
prime radical	C₂³	#432
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

230 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 94 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) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=dh,d^2=i,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^b=fgi,f^c=fgi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42834);