#147143

order	512
nilpotency class	3
isoclinism	isoclinic class of #1902
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#386
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10876
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

517 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54339);