#512007

order	512
nilpotency class	4
isoclinism	isoclinic class of #512006
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#32956
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

910 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=ei,e^2=i,f^2=h,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=fhi,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, 419203);