#147288

order	512
nilpotency class	4
isoclinism	isoclinic class of #12933
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1853
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10908
prime radical	C₂⁴	#137
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

505 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fh,d^2=h,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,g^a=gi,h^a=hi,d^b=dh,e^b=eh,f^b=fi,h^b=hi,d^c=dh,e^c=ehi,f^c=fi,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, 54484);