#490756

order	512
nilpotency class	4
isoclinism	isoclinic class of #490693
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
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

493 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)(3,10)(4,8)(5,14)(6,9)(7,13)(11,16)(12,15)(18,22)(19,25)(20,27)(23,30)(24,31)(26,29)(28,32);(1,3,5,13,6,11,4,12)(2,7,8,10,9,15,14,16);(3,11)(7,15)(10,16)(12,13)(17,18)(19,26)(20,29)(21,23)(22,25)(24,27)(28,31)(30,32);(1,4,6,5)(2,8,9,14)(3,12,11,13)(7,10,15,16)(18,23)(22,30)(24,29)(26,31)

Polycyclic presentation:

2^9|b^2=e,d^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,c^b=cg,d^c=di,f^c=fh,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319599, #72319636, #72319656, …
maximal normal subgroup of	#72319599, #72319636, #72319656, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397952);