#512111

order	512
nilpotency class	4
isoclinism	isoclinic class of #512110
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#33004
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2873 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

935 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=eh,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=gh,h^a=hi,f^b=fi,g^b=gi,e^c=ei,f^c=fhi,g^c=gh,h^c=hi,e^d=ei,f^d=fh,g^d=gh,h^d=hi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419307);