#490931

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)(36,42)(37,40)(38,50)(39,53)(41,47)(43,52)(45,61)(46,60)(48,56)(49,63)(51,57)(54,59)(55,62)(58,64);(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)(34,40)(41,55)(42,53)(43,56)(44,60)(50,63)(57,58)(59,61);(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)(34,41)(36,45)(37,46)(38,51)(39,54)(40,55)(42,57)(43,59)(44,50)(47,52)(48,62)(49,64)(53,58)(56,61)(60,63);(4,16)(8,22)(13,27)(17,26)(20,31)(23,29)(24,32)(28,30)(33,36,37,39)(34,42,40,53)(35,45,46,54)(38,48,49,52)(41,57,55,58)(43,50,56,63)(44,61,60,59)(47,51,62,64)

Polycyclic presentation:

2^9|a^2=g,b^2=egi,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, 398127);