#148856

order	512
nilpotency class	4
isoclinism	isoclinic class of #13472
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2067
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11227
prime radical	C₂⁴	#137
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

473 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=d,d^2=i,e^2=h,f^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,c^b=cf,f^b=fi,e^c=eh,f^c=fi,g^c=gh,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, 56052);