#435250

order	512
nilpotency class	3
isoclinism	isoclinic class of #420416
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15521
prime radical	C₂⁴	C₂²×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

641 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:

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

Polycyclic presentation:

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

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, 342446);