#512610

order	512
nilpotency class	5
isoclinism	isoclinic class of #512609
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂²×#353
commutator	C₄²	C₂³×C₄
Frattini, cosocle	#124	C₂⁴
prime radical	#60651	C₂
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

1072 conjugacy classes of subgroups; only the 192 normal subgroups are shown.

normal subgroups only

This group has 1072 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)(3,15,18,43)(4,10)(5,11)(6,26)(7,13,29,39)(8,14)(9,17,32,50)(12,28,40,25)(16,42,47,55)(19,31,20,44)(21,34)(22,36,53,61)(23,37)(24,38)(27,41,51,33)(30,46,56,64)(35,52,59,54)(45,60,63,57)(48,58,49,62);(1,3)(2,9)(4,16)(5,17)(6,27)(7,18)(8,20)(10,30)(11,31)(12,15)(13,41)(14,33)(19,29)(21,45)(22,47)(23,49)(24,46)(25,50)(26,43)(28,51)(32,39)(34,55)(35,42)(36,60)(37,57)(38,58)(40,44)(48,53)(52,63)(54,64)(56,61)(59,62);(1,4,8,23)(2,10,14,37)(3,16,20,49)(5,24,6,21)(7,22,29,53)(9,30,33,57)(11,38,26,34)(12,35,40,59)(13,36,39,61)(15,42,44,62)(17,46,27,45)(18,47,19,48)(25,54,28,52)(31,58,43,55)(32,56,41,60)(50,64,51,63);(1,5,8,6)(2,11,14,26)(3,17,20,27)(4,21,23,24)(7,25,29,28)(9,31,33,43)(10,34,37,38)(12,39,40,13)(15,32,44,41)(16,45,49,46)(18,50,19,51)(22,52,53,54)(30,55,57,58)(35,36,59,61)(42,60,62,56)(47,63,48,64)

Polycyclic presentation:

2^9|a^2=f,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=eh,f^b=fghi,g^b=gi,d^c=di,f^e=fhi,g^f=gi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419806);