#129685

order	512
nilpotency class	3
isoclinism	isoclinic class of #8359
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1779
prime radical	C₂³×D₄	C₈
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

767 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 767 conjugacy classes of subgroups — too many to draw. Only its 120 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,d^2=gi,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dgi,f^b=fi,f^c=fg

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, 36881);