#496033

order	512
nilpotency class	3
isoclinism	isoclinic class of #2031
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₁₆	#124
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#21635
prime radical	#575	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆ ◁ C₂²×C₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

469 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=e,d^2=g,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,f^a=fi,c^b=ch,e^b=ei,f^c=fi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224962, #72320755
maximal normal subgroup of	#72224962, #72320755

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403229);