#503558

order	512
nilpotency class	4
isoclinism	isoclinic class of #501256
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 ◁ C₂×#446 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

751 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=ei,b^2=e,c^2=efh,d^2=e,f^2=hi,g^2=hi,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=gh,h^b=hi,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410754);