#413728

order	512
nilpotency class	3
isoclinism	isoclinic class of #402003
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2494
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	C₄⋊#401
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

597 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)(3,13,16,8)(4,9,20,32)(5,22)(6,25)(7,11)(10,23)(12,26)(14,30)(15,39,40,50)(17,29,42,36)(18,27)(19,45,46,64)(21,49,48,33)(24,51)(28,41,35,59)(31,57,58,47)(34,52)(37,60)(38,54)(43,62)(44,56)(53,63)(55,61);(1,3)(2,8)(4,18,20,14)(5,15)(6,17)(7,16)(9,30,32,27)(10,28)(11,13)(12,29)(19,43,46,37)(21,44,48,38)(22,50)(23,41)(24,40)(25,36)(26,42)(31,55,58,53)(33,56,49,54)(34,35)(39,51)(45,60,64,62)(47,63,57,61)(52,59);(1,4)(2,9,11,32)(3,14)(5,19)(6,21)(7,20)(8,27,13,30)(10,31,34,58)(12,33,25,49)(15,37)(16,18)(17,38)(22,45,51,64)(23,47)(24,46)(26,48)(28,53,35,55)(29,54,36,56)(39,60,50,62)(40,43)(41,61)(42,44)(52,57)(59,63);(1,5,6,23,7,24,26,52)(2,10,12,22,11,34,25,51)(3,15,17,41,16,40,42,59)(4,19,21,47,20,46,48,57)(8,28,29,50,13,35,36,39)(9,31,33,45,32,58,49,64)(14,37,38,61,18,43,44,63)(27,53,54,62,30,55,56,60)

Polycyclic presentation:

2^9|a^2=efg,b^2=ef,c^2=e,d^2=h,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,g^b=gi,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, 320924);