#487172

order	512
nilpotency class	4
isoclinism	isoclinic class of #486787
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
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 903 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) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=hi,e^b=eg,f^b=fhi,d^c=dg,e^c=eh,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, 394368);