#507427

order	512
nilpotency class	4
isoclinism	isoclinic class of #501766
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#476
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3147	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

540 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=efg,b^2=eg,d^2=e,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,d^b=di,f^c=fh,g^c=gi,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 414623);