#512559

order	512
nilpotency class	3
isoclinism	isoclinic class of #2883
automorphism	?
homology	Abelianization (H₁) = C₂³×C₁₆ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	C₂²×C₂²⋊C₄
commutator	C₂²	C₂³×C₁₆
Frattini, cosocle	C₂×C₁₆	C₂⁴
socle	C₂	C₂²×C₂²⋊C₁₆
prime radical	C₂²×#41	C₈
LCS	G ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂×#504 ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

763 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^9|a^2=f,f^2=g,g^2=h,h^2=i,b^a=be,e^a=ei,f^b=fi,d^c=di

Conjugacy classes & character table

164 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419755);