#512495

order	512
nilpotency class	4
isoclinism	isoclinic class of #512493
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#33272
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1747 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

358 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 120 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) = 64

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

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

Polycyclic presentation:

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

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, 419691);