#506468

order	512
nilpotency class	4
isoclinism	isoclinic class of #500592
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3253
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#28280
prime radical	#3147	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

526 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)(3,17)(4,24,28,57)(5,29)(6,15)(7,33)(8,13)(9,16)(10,20)(11,26,44,61)(12,31)(14,32)(18,48,51,55)(19,45)(21,38)(22,40)(23,39)(25,58,54,63)(27,46,35,56)(30,37)(34,43,60,42)(36,59,64,53)(41,49,47,50)(52,62);(1,3)(2,10,16,40)(4,25,28,54)(5,30)(6,20)(7,21)(8,22)(9,23)(11,41)(12,45,33,37)(13,17,15,39)(14,38,29,62)(18,27,51,35)(19,32)(24,55)(26,59,61,53)(31,52)(34,50,60,49)(36,42)(43,64)(44,47)(46,63)(48,57)(56,58);(1,4)(2,11)(3,18)(5,26)(6,27)(7,34)(8,35)(9,28)(10,36)(12,24)(13,42)(14,46)(15,43)(16,44)(17,47)(19,49)(20,25)(21,53)(22,54)(23,51)(29,56)(30,50)(31,60)(32,61)(33,57)(37,63)(38,55)(39,41)(40,64)(45,58)(48,62)(52,59);(1,5,6,31,9,32,8,7)(2,12,13,29,16,33,15,14)(3,19,20,21,23,30,22,52)(4,26,27,60,28,61,35,34)(10,37,17,38,40,45,39,62)(11,24,42,56,44,57,43,46)(18,49,25,53,51,50,54,59)(36,63,47,55,64,58,41,48)

Polycyclic presentation:

2^9|a^2=eh,b^2=eg,d^2=e,e^2=i,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,h^a=hi,c^b=cg,d^b=di,f^c=fhi,g^c=gi,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413664);