#416780

order	512
nilpotency class	3
isoclinism	isoclinic class of #398772
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2439
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15231
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

693 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 323976);