#490941

order	512
nilpotency class	4
isoclinism	isoclinic class of #490891
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	C₂³	#459
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

404 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 128 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398137);