#148935

order	512
nilpotency class	3
isoclinism	isoclinic class of #2081
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#397
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11255
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

451 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56131);