#418163

order	512
nilpotency class	3
isoclinism	isoclinic class of #417579
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60645	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1607 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:

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

Polycyclic presentation:

2^9|b^2=e,c^2=e,d^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,f^c=fi,h^d=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, 325359);