#459884

order	512
nilpotency class	3
isoclinism	isoclinic class of #454109
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2670
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16351
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

895 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)(3,14,18,40)(4,10)(5,25,28,11)(6,12)(7,30)(8,13)(9,17,33,19)(15,43,46,34)(16,42)(20,51,45,31)(21,35,52,61)(22,36)(23,54)(24,37)(26,38,53,56)(27,41,29,39)(32,48)(44,60)(47,57)(49,55)(50,59)(58,64)(62,63);(1,3,6,19,8,18,7,17)(2,9,12,40,13,33,30,14)(4,20,22,46,24,45,23,15)(5,16,29,48,28,50,27,49)(10,34,36,31,37,43,54,51)(11,32,41,59,25,55,39,42)(21,47,53,64,52,63,26,44)(35,60,56,57,61,58,38,62);(1,4,6,22,8,24,7,23)(2,10,12,36,13,37,30,54)(3,15,17,45,18,46,19,20)(5,26,27,52,28,53,29,21)(9,31,14,34,33,51,40,43)(11,38,39,61,25,56,41,35)(16,47,48,64,50,63,49,44)(32,58,59,62,55,60,42,57);(1,5,6,29,8,28,7,27)(2,11,12,41,13,25,30,39)(3,16,19,48,18,50,17,49)(4,21,22,53,24,52,23,26)(9,32,40,59,33,55,14,42)(10,35,36,56,37,61,54,38)(15,44,20,47,46,64,45,63)(31,57,43,58,51,62,34,60)

Polycyclic presentation:

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

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, 367080);