#491629

order	512
nilpotency class	4
isoclinism	isoclinic class of #491572
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#493
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#33773	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

561 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)(3,9)(4,12)(5,7)(6,11)(8,13)(10,15)(14,16)(17,18)(19,27)(20,24)(21,30)(22,32)(23,26)(25,28)(29,31);(1,3,4,11,5,10,13,16)(2,6,8,9,7,14,12,15);(2,7)(6,14)(8,12)(9,15)(17,19,28,26,22,29,31,32,23,25,27,18,20,30,21,24);(3,10)(6,14)(9,15)(11,16)(17,20,23,22)(18,24,26,32)(19,29,25,30)(21,27,31,28)

Polycyclic presentation:

2^9|b^2=e,c^2=fgh,d^2=i,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=dg,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398825);