#488180

order	512
nilpotency class	4
isoclinism	isoclinic class of #487699
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	#54852	C₂
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 899 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) = 32

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 395376);