#503561

order	512
nilpotency class	4
isoclinism	isoclinic class of #501244
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

744 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=ei,b^2=e,c^2=efh,d^2=e,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=ghi,h^a=hi,c^b=cg,f^b=fhi,g^b=ghi,h^b=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, 410757);