#507596

order	512
nilpotency class	4
isoclinism	isoclinic class of #500306
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	#29273
prime radical	C₂³	#575
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 395 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)(3,17)(4,24,28,42)(5,29)(6,16)(7,32)(8,13)(9,15)(10,20)(11,27,44,25)(12,30)(14,31)(18,50,53,64)(19,49)(21,38)(22,39)(23,40)(26,59,35,46)(33,45)(34,43,58,47)(36,55,41,61)(37,54)(48,51,62,60)(52,57,56,63);(1,3)(2,10,15,39)(4,18)(5,19)(6,20)(7,33)(8,22)(9,23)(11,41,44,36)(12,37,32,49)(13,17,16,40)(14,45,29,38)(21,30)(24,57,42,63)(25,51)(26,52)(27,60)(28,53)(31,54)(34,61)(35,56)(43,62,47,48)(46,50,59,64)(55,58);(1,4)(2,11,15,44)(3,18)(5,25)(6,26)(7,34)(8,35)(9,28)(10,36,39,41)(12,42,32,24)(13,43,16,47)(14,46,29,59)(17,48,40,62)(19,51)(20,52)(21,55)(22,56)(23,53)(27,31)(30,58)(33,61)(37,57,49,63)(38,50,45,64)(54,60);(1,5,6,30,9,31,8,7)(2,12,13,29,15,32,16,14)(3,19,20,21,23,54,22,33)(4,25,26,58,28,27,35,34)(10,37,17,38,39,49,40,45)(11,42,43,59,44,24,47,46)(18,51,52,55,53,60,56,61)(36,57,48,50,41,63,62,64)

Polycyclic presentation:

2^9|a^2=eh,b^2=g,c^2=g,d^2=e,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,f^b=fi,f^c=fh,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, 414792);