#492499

order	512
nilpotency class	4
isoclinism	isoclinic class of #492252
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#33964	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

431 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,d^2=g,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,c^b=cgi,d^b=de,e^b=eg,f^c=fh,h^c=hi,e^d=eg

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320666
maximal normal subgroup of	#72320666

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 399695);