#490410

order	512
nilpotency class	4
isoclinism	isoclinic class of #490121
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#62383	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

624 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=gh,e^2=gi,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=dghi,e^a=eg,f^a=fhi,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,e^c=eh,f^c=fhi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397606);