#491058

order	512
nilpotency class	4
isoclinism	isoclinic class of #491017
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#3169	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

441 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398254);