#148768

order	512
nilpotency class	4
isoclinism	isoclinic class of #140091
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1918
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11219
prime radical	C₂⁴	C₂×Q₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

463 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=d,d^2=i,e^2=h,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,f^a=fh,h^a=hi,c^b=cf,e^b=eh,f^b=fh,h^b=hi,e^c=eh,f^c=fh,g^c=gi,h^c=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, 55964);