#487067

order	512
nilpotency class	4
isoclinism	isoclinic class of #486811
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×C₂²⋊C₄	C₂⁴
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1030 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 1030 conjugacy classes of subgroups — too many to draw. Only its 116 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 — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=g,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=hi,d^b=di,e^b=eg,f^b=fhi,d^c=dgi,e^c=eh,f^e=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 394263);