#507570

order	512
nilpotency class	4
isoclinism	isoclinic class of #499952
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3205
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#29254
prime radical	#3147	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

497 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 497 conjugacy classes of subgroups — too many to draw. Only its 142 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)(3,15)(4,21)(5,13)(6,29)(7,31)(8,14)(9,18)(10,24)(11,27)(12,30)(16,47)(17,52)(19,42)(20,46)(22,40)(23,58)(25,43)(26,57)(28,44)(32,36)(33,41)(34,55)(35,53)(37,54)(38,61)(39,59)(45,48)(49,56)(50,62)(51,64)(60,63);(1,3)(2,9,14,37)(4,16)(5,17)(6,18)(7,32)(8,20)(10,38,40,34)(11,35,31,52)(12,15,29,46)(13,42,44,36)(19,27)(21,56,57,62)(22,48)(23,49)(24,60)(25,50)(26,51)(28,53)(30,54)(33,61)(39,63,41,45)(43,47,58,64)(55,59);(1,4)(2,10,14,40)(3,16)(5,22)(6,23)(7,33)(8,26)(9,34,37,38)(11,21,31,57)(12,39,29,41)(13,43,44,58)(15,45,46,63)(17,48)(18,49)(19,55)(20,51)(24,28)(25,30)(27,59)(32,61)(35,62,52,56)(36,47,42,64)(50,54)(53,60);(1,5,6,27,8,28,30,7)(2,11,12,13,14,31,29,44)(3,17,18,19,20,53,54,32)(4,22,23,59,26,24,25,33)(9,35,15,42,37,52,46,36)(10,21,39,43,40,57,41,58)(16,48,49,55,51,60,50,61)(34,62,45,64,38,56,63,47)

Polycyclic presentation:

2^9|b^2=g,c^2=g,d^2=e,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,g^a=gi,h^a=hi,c^b=cg,f^b=fi,f^c=fhi,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, 414766);