#504342

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	#26533
prime radical	#3147	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

431 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:

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

Polycyclic presentation:

2^9|a^2=efh,b^2=fh,c^2=efh,d^2=e,e^2=i,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,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, 411538);