#355278

order	512
nilpotency class	3
isoclinism	isoclinic class of #22284
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2328
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#13779
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

829 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 262474);