#501051

order	512
nilpotency class	4
isoclinism	isoclinic class of #501046
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3050	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1162 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)(3,4)(9,10,15,29,27,17,26,36,14,20,13,18,12,19,31,34)(11,21,38,32,22,40,28,16,25,39,23,37,24,35,30,33);(2,4)(5,6)(7,8);(6,8)(9,11)(10,16)(12,22)(13,28)(14,25)(15,30)(17,32)(18,35)(19,37)(20,33)(21,36)(23,26)(24,27)(29,40)(31,38)(34,39);(2,4)(9,12,14,27)(10,17,20,19)(11,22,25,24)(13,26,15,31)(16,32,33,37)(18,34,29,36)(21,35,39,40)(23,30,38,28)

Polycyclic presentation:

2^9|a^2=fi,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, 408247);