#490554

order	512
nilpotency class	4
isoclinism	isoclinic class of #12463
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1675
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂	C₂×#1675
prime radical	C₂×#135	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

385 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397750);