#486470

order	512
nilpotency class	4
isoclinism	isoclinic class of #486127
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

764 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 134 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 — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=i,d^2=g,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fi,h^a=hi,c^b=ci,d^b=dg,e^b=eg,f^b=fhi,e^c=eh,f^c=fi,f^e=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393666);