#359420

order	512
nilpotency class	3
isoclinism	isoclinic class of #2104
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#401
commutator, prime radical	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	C₄×#401
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

655 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,c^2=e,d^2=efgh,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,c^b=cg,g^b=gi,h^b=hi,f^c=fi,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266616);