#491205

order	512
nilpotency class	4
isoclinism	isoclinic class of #491109
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
prime radical	#19269	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×#41 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

453 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)(3,11)(4,10)(5,7)(6,9)(8,12)(13,16)(14,15)(17,18)(19,27)(20,24)(21,30)(22,32)(23,26)(25,28)(29,31);(2,7)(8,15)(9,10)(11,16)(17,19,28,26,22,29,31,32,23,25,27,18,20,30,21,24);(1,3)(2,8)(4,12)(5,13)(6,14)(7,15)(9,11)(10,16);(1,4,5,6)(2,9,7,10)(3,12,13,14)(8,11,15,16)(17,20,23,22)(18,24,26,32)(19,29,25,30)(21,27,31,28)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321876
maximal normal subgroup of	#72321876

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398401);