#402150

order	512
nilpotency class	3
isoclinism	isoclinic class of #396106
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2459
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14834
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

759 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:

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

Polycyclic presentation:

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

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, 309346);