#238337

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

704 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=eh,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, 145533);