#512641

order	512
nilpotency class	5
isoclinism	isoclinic class of #512640
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂²×#354
commutator	C₄²	C₂³×C₄
Frattini, cosocle	#124	C₂⁴
prime radical	#3500	C₂²
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×Q₈ ◁ C₂×#41 ◁ #3500 ◁ G
LECS	G ▷ #124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

797 conjugacy classes of subgroups; only the 192 normal subgroups are shown.

normal subgroups only

This group has 797 conjugacy classes of subgroups — too many to draw. Only its 192 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)(3,15,19,31)(4,10)(5,11)(6,27)(7,13,29,39)(8,14)(9,18,32,51)(12,28,41,26)(16,43,46,58)(17,42,50,33)(20,44,21,40)(22,34)(23,36,53,62)(24,37)(25,38)(30,45,56,63)(35,52,59,54)(47,55,48,60)(49,61,64,57);(1,3,8,21)(2,9)(4,16,24,48)(5,17,6,18)(7,19,29,20)(10,30)(11,31)(12,40)(13,42)(14,33)(15,41)(22,45,25,49)(23,46,53,47)(26,50,28,51)(27,44)(32,39)(34,55)(35,60)(36,61)(37,57)(38,58)(43,59)(52,63,54,64)(56,62);(1,4,8,24)(2,10,14,37)(3,16,21,48)(5,25,6,22)(7,23,29,53)(9,30,33,57)(11,38,27,34)(12,35,41,59)(13,36,39,62)(15,43,40,60)(17,49,18,45)(19,46,20,47)(26,54,28,52)(31,58,44,55)(32,56,42,61)(50,64,51,63);(1,5,8,6)(2,11,14,27)(3,17,21,18)(4,22,24,25)(7,26,29,28)(9,31,33,44)(10,34,37,38)(12,39,41,13)(15,42,40,32)(16,45,48,49)(19,50,20,51)(23,52,53,54)(30,55,57,58)(35,36,59,62)(43,56,60,61)(46,63,47,64)

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=i,d^2=i,e^2=h,g^2=i,b^a=be,e^a=eg,g^a=gh,h^a=hi,e^b=ehi,f^b=fghi,g^b=gi,d^c=di,f^e=fhi,g^f=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, 419837);