#491368

order	512
nilpotency class	4
isoclinism	isoclinic class of #491233
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×#41 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

481 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398564);