#238338

order	512
nilpotency class	3
isoclinism	isoclinic class of #238331
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#393
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

708 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=i,d^2=hi,f^2=hi,g^2=h,b^a=be,c^a=cf,f^a=fi,g^a=ghi,c^b=cg,d^b=dh,f^b=fi,d^c=di,e^c=eh,f^c=fhi,g^c=gh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 145534);