#494410

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	C₂×#575	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1265 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341733
maximal normal subgroup of	#72341733

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401606);