#460708

order	512
nilpotency class	3
isoclinism	isoclinic class of #455830
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2648
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16370
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

889 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:

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

Polycyclic presentation:

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

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, 367904);