#494809

order	512
nilpotency class	4
isoclinism	isoclinic class of #33657
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#481
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1919
prime radical	C₂⁴	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

575 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=eh,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=di,f^b=fi,e^c=ei,f^c=fhi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402005);