#492879

order	512
nilpotency class	4
isoclinism	isoclinic class of #33388
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂²×#102
commutator	C₈	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂²×#340
prime radical	C₂²×#41	D₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1541 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400075);