#403799

order	512
nilpotency class	3
isoclinism	isoclinic class of #394813
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2436
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14882
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

924 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 310995);