#510639

order	512
nilpotency class	4
isoclinism	isoclinic class of #510638
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#31796
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#525	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2930 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

429 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)(3,14,18,9)(4,19)(5,24)(6,26)(7,12)(8,11)(10,22)(13,29)(15,46,47,62)(16,34,31,36)(17,35,51,45)(20,52,54,64)(21,39)(23,43)(25,41)(27,42,59,58)(28,60)(30,57)(32,40)(33,48,44,49)(37,50,63,55)(38,61)(53,56);(1,3,5,16,8,18,7,31)(2,9,24,36,11,14,12,34)(4,20,32,50,21,54,22,55)(6,27,25,51,29,59,30,17)(10,37,19,52,40,63,39,64)(13,42,57,45,26,58,41,35)(15,28,49,23,47,61,48,56)(33,43,62,38,44,53,46,60);(1,4,5,22,8,21,7,32)(2,10,12,19,11,40,24,39)(3,15,31,49,18,47,16,48)(6,28,25,56,29,61,30,23)(9,33,36,46,14,44,34,62)(13,43,41,38,26,53,57,60)(17,50,59,54,51,55,27,20)(35,52,42,37,45,64,58,63);(6,29)(9,14)(13,26)(17,51)(20,54)(23,56)(25,30)(27,59)(28,61)(33,44)(34,36)(38,60)(41,57)(43,53)(46,62)(50,55)

Polycyclic presentation:

2^9|a^2=efh,b^2=ef,c^2=ehi,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=defh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,g^b=gh,h^b=hi,e^c=ei,g^c=gh,h^c=hi,g^d=gi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 417835);