#507673

order	512
nilpotency class	4
isoclinism	isoclinic class of #501046
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

597 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fg,b^2=egi,c^2=g,d^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,f^c=fh,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, 414869);