#486919

order	512
nilpotency class	4
isoclinism	isoclinic class of #486787
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

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	#2403	C₂²
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 934 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) = 32

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

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

Polycyclic presentation:

2^9|a^2=g,d^2=i,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, 394115);