#503824

order	512
nilpotency class	4
isoclinism	isoclinic class of #500736
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3325
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#26111
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

401 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=e,c^2=efh,d^2=e,e^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,g^b=gh,h^b=hi,f^c=fi,g^c=gh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411020);