#511907

order	512
nilpotency class	4
isoclinism	isoclinic class of #511905
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#32902
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3584	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1030 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ei,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fh,g^a=gi,h^a=hi,f^c=fh,h^c=hi,g^d=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419103);