#494418

order	512
nilpotency class	3
isoclinism	isoclinic class of #3019
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	C₂×#120
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1869
prime radical	#3449	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1277 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:

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

Polycyclic presentation:

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

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, 401614);