#493074

order	512
nilpotency class	4
isoclinism	isoclinic class of #33388
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂²×#102
commutator	C₈	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×C₄×#102
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×M₁₆ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1121 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,e^2=hi,f^2=i,g^2=h,h^2=i,b^a=be,e^a=eh,g^a=gh,h^a=hi,e^b=ei,d^c=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285581, #72285693, #72285871, …
maximal normal subgroup of	#72285581, #72285693, #72285871, …
maximal quotient of	#72310117, #72321241, M₁₆×#360, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400270);