#497180

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

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

442 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)(3,6)(4,8)(5,7)(9,10)(11,13)(12,15)(14,16)(17,18,23,28,24,19,29,32,22,27,37,43,21,31,25,38)(20,26,35,42,33,30,44,45,34,41,46,48,36,40,39,47);(1,3,5,4)(2,6,7,8)(9,11,13,10)(12,15,16,14)(17,19,23,32,24,27,29,43,22,31,37,38,21,18,25,28)(20,30,35,45,33,41,44,48,34,40,46,47,36,26,39,42);(1,4,5,3)(2,6,7,8)(10,11)(14,15)(17,20,22,34)(18,26,27,41)(19,30,31,40)(21,36,24,33)(23,35,37,46)(25,39,29,44)(28,42,43,48)(32,45,38,47);(9,12,13,16)(10,14,11,15)(17,21,22,24)(18,19,27,31)(20,33,34,36)(23,25,37,29)(26,40,41,30)(28,32,43,38)(35,44,46,39)(42,47,48,45)

Polycyclic presentation:

2^9|a^2=g,b^2=fg,c^2=ef,d^2=e,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cei,d^b=di,d^c=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404376);