#152206

order	512
nilpotency class	3
isoclinism	isoclinic class of #471
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₁₆	#121
commutator	C₂×C₄	C₂²×C₁₆
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂	#12783
prime radical	C₂×#41	C₂×C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆ ◁ C₂²×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

303 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59402);