#487016

order	512
nilpotency class	4
isoclinism	isoclinic class of #486883
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	#52712	C₂
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 801 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=i,e^2=g,f^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fi,h^a=hi,c^b=ci,e^b=eg,f^b=fhi,d^c=dg,e^c=eh,f^c=fi,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, 394212);