#512493

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×D₁₆	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1747 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 363 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)(5,15)(6,24)(9,29)(10,35)(11,37)(16,44)(17,49)(18,41)(20,27)(22,23)(25,43)(26,32)(28,30)(31,58)(33,57)(34,56)(38,46)(39,40)(42,50)(45,55)(47,64)(48,63)(51,61)(53,62)(59,60);(1,2,6,13,7,8,24,12)(3,9,17,34,18,26,39,33)(4,20,19,10,23,35,22,27)(5,11,15,30,25,28,43,37)(14,32,40,56,41,29,49,57)(16,45,46,31,48,60,47,53)(21,51,50,36,52,61,42,54)(38,62,44,59,64,58,63,55);(1,3,15,40,6,17,25,41,7,18,43,49,24,39,5,14)(2,9,30,56,13,34,28,29,8,26,37,57,12,33,11,32)(4,16,42,63,22,47,52,64,23,48,50,44,19,46,21,38)(10,31,51,62,20,45,54,55,27,53,61,58,35,60,36,59);(1,4)(2,10,8,27)(3,16)(5,21)(6,22)(7,23)(9,31,26,53)(11,36,28,54)(12,35,13,20)(14,38)(15,42)(17,47)(18,48)(19,24)(25,52)(29,55,32,59)(30,51,37,61)(33,60,34,45)(39,46)(40,63)(41,64)(43,50)(44,49)(56,62,57,58)

Polycyclic presentation:

2^9|b^2=eg,c^2=fh,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, 419689);