#507767

order	512
nilpotency class	4
isoclinism	isoclinic class of #500307
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3227
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#29392
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

415 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 415 conjugacy classes of subgroups — too many to draw. Only its 124 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=e,d^2=g,e^2=i,f^2=h,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,d^b=di,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, 414963);