#147275

order	512
nilpotency class	4
isoclinism	isoclinic class of #13030
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1879
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10907
prime radical	C₂⁴	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

509 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=hi,e^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,h^a=hi,d^b=dhi,e^b=ehi,h^b=hi,d^c=dhi,e^c=eh,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, 54471);