#490930

order	512
nilpotency class	4
isoclinism	isoclinic class of #490873
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	#466	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

419 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2,6,10)(3,11,15,25)(4,8,16,22)(5,18,12,9)(7,19,21,14)(13,24,27,32)(17,29,26,23)(20,30,31,28)(33,34)(35,44)(36,41)(37,49)(38,53)(39,43)(40,46)(42,51)(45,60)(47,61)(48,59)(50,55)(52,63)(54,58)(56,64)(57,62);(1,3,12,19,6,15,5,14)(2,7,18,25,10,21,9,11)(4,13,26,30,16,27,17,28)(8,20,29,32,22,31,23,24);(1,4)(2,8)(3,13)(5,17)(6,16)(7,20)(9,23)(10,22)(11,24)(12,26)(14,28)(15,27)(18,29)(19,30)(21,31)(25,32)(33,35)(34,40)(36,45)(37,50)(38,54)(39,47)(41,55)(42,58)(43,57)(44,49)(46,51)(48,62)(52,64)(53,56)(59,60)(61,63);(4,16)(8,22)(13,27)(17,26)(20,31)(23,29)(24,32)(28,30)(33,36,39,38)(34,41,43,53)(35,45,47,54)(37,48,52,51)(40,55,57,56)(42,49,59,63)(44,60,61,58)(46,50,62,64)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=i,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^c=dg,f^c=fh,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398126);