#490828

order	512
nilpotency class	4
isoclinism	isoclinic class of #490777
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	#33660	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

489 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)(3,9,10,16)(4,12,13,7)(6,14,15,11)(17,18,19,22)(20,23,26,30)(21,29,27,25)(24,32,31,28);(1,3,4,11,5,10,13,14)(2,6,7,9,8,15,12,16)(18,22)(23,30)(24,31)(25,29);(18,23)(20,28)(21,27)(22,30)(24,25)(26,32)(29,31);(3,10)(6,15)(9,16)(11,14)

Polycyclic presentation:

2^9|a^2=gi,b^2=ei,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,f^c=fh,h^c=hi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319626
maximal normal subgroup of	#72319626

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398024);