#238396

order	512
nilpotency class	3
isoclinism	isoclinic class of #238391
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	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

718 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:

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

Polycyclic presentation:

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

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, 145592);