#487280

order	512
nilpotency class	4
isoclinism	isoclinic class of #486891
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	#43882	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

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

normal subgroups only

This group has 798 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)(3,7)(4,9)(5,8)(6,11)(10,14)(12,16)(13,15)(17,18,23,28)(19,27,22,31)(20,25,30,32)(21,24,29,26);(1,3,6,4)(2,7,11,9)(5,13,14,12)(8,16,10,15)(18,24)(19,22)(21,29)(25,27)(26,28)(31,32);(2,8)(3,4)(5,14)(7,15)(9,16)(10,11)(17,19)(20,29)(21,30)(22,23)(24,26)(27,31);(1,4,6,3)(2,9,11,7)(5,13,14,12)(8,15,10,16)(17,20,23,30)(18,25,28,32)(19,29,22,21)(24,27,26,31)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,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=dgi,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, 394476);