#507506

order	512
nilpotency class	4
isoclinism	isoclinic class of #501064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

484 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,5,8)(3,7,4,6)(9,10,12,14)(11,13,15,16)(17,18)(19,28)(20,25)(21,36)(22,34)(23,27)(24,31)(26,29)(30,41)(32,47)(33,43)(35,40)(37,48)(38,45)(39,46)(42,44);(1,3,5,4)(2,6,8,7)(9,11,12,15)(10,13,14,16)(17,18)(19,24)(20,34)(21,26)(22,25)(23,27)(28,31)(29,36)(30,42)(32,38)(33,39)(35,48)(37,40)(41,44)(43,46)(45,47);(11,15)(13,16)(17,19,29,44,22,32,37,46,23,33,35,45,20,30,21,31)(18,24,36,41,25,38,40,43,27,39,48,47,34,42,26,28);(1,4,5,3)(2,7,8,6)(17,20,23,22)(18,25,27,34)(19,30,33,32)(21,35,37,29)(24,38,39,42)(26,36,40,48)(28,41,43,47)(31,45,46,44)

Polycyclic presentation:

2^9|a^2=eg,b^2=eg,c^2=fh,d^2=ei,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cg,d^b=di

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, 414702);