#485976

order	512
nilpotency class	4
isoclinism	isoclinic class of #485663
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	#23587	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

867 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 867 conjugacy classes of subgroups — too many to draw. Only its 120 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)(3,10)(4,8)(5,14)(6,9)(7,12)(11,15)(13,16)(17,18)(19,22)(20,23)(21,25)(24,29)(26,31)(27,32)(28,30);(1,3,6,13)(2,7,9,15)(4,12,5,11)(8,16,14,10)(17,19)(18,22)(20,28)(21,26)(23,32)(24,30)(25,31)(27,29);(18,23)(19,26)(20,29)(22,30)(24,25)(31,32);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,10,15,16)(19,26)(22,31)(27,28)(30,32)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393172);