#490003

order	512
nilpotency class	4
isoclinism	isoclinic class of #489691
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	D₈×#41	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×#41 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

846 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:

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

Polycyclic presentation:

2^9|a^2=h,d^2=h,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fh,g^a=gi,e^b=eg,f^b=fh,g^b=gi,e^c=eh,f^c=fh

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397199);