#490517

order	512
nilpotency class	4
isoclinism	isoclinic class of #490428
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	#33669	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

507 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)(3,10)(4,8)(5,14)(6,9)(7,12)(11,15)(13,16)(17,18)(19,24)(20,23)(21,30)(22,26)(25,28)(27,31)(29,32);(1,3,5,12,6,13,4,11)(2,7,8,10,9,15,14,16);(18,23)(19,21)(20,28)(24,25)(26,32)(27,29)(30,31);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,10,15,16)(17,19,22,21)(18,24,26,30)(20,27,29,28)(23,31,32,25)

Polycyclic presentation:

2^9|b^2=e,d^2=gi,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^c=di,f^c=fh,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285417, #72285445, #72285618, …
maximal normal subgroup of	#72285417, #72285445, #72285618, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397713);