#492150

order	512
nilpotency class	4
isoclinism	isoclinic class of #491760
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#3391	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

424 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=gi,c^2=i,d^2=g,e^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,h^a=hi,d^b=dei,e^b=eg,d^c=di,f^c=fh,h^c=hi,e^d=eg

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, 399346);