#469167

order	512
nilpotency class	3
isoclinism	isoclinic class of #455563
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2730
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16648
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

800 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)(4,10,21,37)(5,22,25,41)(6,27)(7,12)(8,13)(9,17)(11,26,40,24)(15,42,46,59)(16,48,51,31)(18,33)(19,53)(20,38,28,36)(23,57)(29,32)(30,45,34,47)(35,55)(39,54)(43,50,56,52)(44,61)(49,60)(58,64)(62,63);(1,3,6,17,8,18,7,29)(2,9,12,14,13,32,27,33)(4,15,28,47,21,46,20,45)(5,16,26,52,25,51,24,50)(10,34,38,59,37,30,36,42)(11,31,41,43,40,48,22,56)(19,44,23,49,54,64,55,63)(35,61,39,62,57,58,53,60);(1,4)(2,10,13,37)(3,15)(5,23)(6,28)(7,20)(8,21)(9,30,32,34)(11,39,40,53)(12,38,27,36)(14,42,33,59)(16,49)(17,47)(18,46)(19,24)(22,35,41,57)(25,55)(26,54)(29,45)(31,60,48,62)(43,61,56,58)(44,50)(51,63)(52,64);(1,5,6,26,8,25,7,24)(2,11,27,22,13,40,12,41)(3,16,17,52,18,51,29,50)(4,19,28,23,21,54,20,55)(9,31,33,56,32,48,14,43)(10,35,36,53,37,57,38,39)(15,44,47,49,46,64,45,63)(30,58,59,62,34,61,42,60)

Polycyclic presentation:

2^9|a^2=fg,b^2=efg,c^2=g,d^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,h^b=hi,d^c=de,e^c=ei,f^c=fi,e^d=ei,f^d=fi,h^d=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 376363);