#491880

order	512
nilpotency class	4
isoclinism	isoclinic class of #491728
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	D₄×D₈	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×SD₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

446 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319599
maximal normal subgroup of	#72319599

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 399076);