#405890

order	512
nilpotency class	3
isoclinism	isoclinic class of #395587
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2411
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14920
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

965 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:

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

Polycyclic presentation:

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

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, 313086);