#238336

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

712 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)(4,18)(5,23)(9,33,24,27)(10,28,37,47)(12,32,30,19)(14,42,38,54)(15,45)(16,49)(21,55)(22,52)(25,26)(31,35,53,57)(34,59,36,58)(39,46)(40,50)(43,60,44,61)(48,62)(51,56)(63,64);(2,9,11,24)(3,13)(4,18)(8,27,29,33)(10,36,37,34)(12,35,30,57)(14,43,38,44)(15,46)(16,40)(17,41)(19,53,32,31)(21,51)(22,52)(28,59,47,58)(39,45)(42,61,54,60)(48,64)(49,50)(55,56)(62,63);(1,2,7,12)(3,14,17,10)(4,19,18,29)(5,24,26,57)(6,11,20,30)(8,22,32,52)(9,23,35,25)(13,38,41,37)(15,47,45,54)(16,43,50,36)(21,27,51,31)(28,46,42,39)(33,56,53,55)(34,40,44,49)(48,60,63,58)(59,62,61,64);(1,3,7,17)(2,10,12,14)(4,15,18,45)(5,16,26,50)(6,13,20,41)(8,28,32,42)(9,34,35,44)(11,37,30,38)(19,54,29,47)(21,48,51,63)(22,39,52,46)(23,49,25,40)(24,36,57,43)(27,58,31,60)(33,59,53,61)(55,62,56,64)

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=i,d^2=i,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, 145532);