#413719

order	512
nilpotency class	3
isoclinism	isoclinic class of #412862
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2858
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15153
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

703 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:

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

Polycyclic presentation:

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

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, 320915);