#500255

order	512
nilpotency class	4
isoclinism	isoclinic class of #500250
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24058
commutator, Frattini, cosocle	C₂×#41	C₂⁴
prime radical	#3486	C₂²
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×2₋¹⁺⁴ ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 813 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=e,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=ghi,c^b=cg,e^b=ei,f^b=fh,d^c=dh,f^c=fi,g^c=gi,h^c=hi,e^d=ei,g^f=gi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407451);