#453884

order	512
nilpotency class	3
isoclinism	isoclinic class of #427666
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2527
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16197
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

531 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)(3,9,17,32)(4,19)(5,11)(6,14)(7,13)(8,12)(10,23)(15,33,43,58)(16,30,45,37)(18,41,28,31)(20,29,46,48)(21,51)(22,35)(24,36)(25,40)(26,39)(27,38)(34,53)(42,63,64,60)(44,59,47,57)(49,55)(50,62,52,56)(54,61);(1,3,6,18,8,17,7,28)(2,9,13,31,12,32,14,41)(4,20,24,15,22,46,23,43)(5,16,26,44,25,45,27,47)(10,33,35,29,36,58,19,48)(11,37,38,59,40,30,39,57)(21,52,54,42,55,50,53,64)(34,60,49,62,61,63,51,56);(1,4,7,23,8,22,6,24)(2,10,13,35,12,36,14,19)(3,15,28,20,17,43,18,46)(5,21,26,54,25,55,27,53)(9,29,31,58,32,48,41,33)(11,34,39,51,40,61,38,49)(16,42,44,50,45,64,47,52)(30,56,59,63,37,62,57,60);(1,5,6,27,8,25,7,26)(2,11,14,38,12,40,13,39)(3,16,28,44,17,45,18,47)(4,21,24,53,22,55,23,54)(9,30,31,59,32,37,41,57)(10,34,19,49,36,61,35,51)(15,42,20,50,43,64,46,52)(29,56,58,63,48,62,33,60)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 361080);