#449835

order	512
nilpotency class	3
isoclinism	isoclinic class of #417700
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1316 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=ei,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,g^b=gi,f^c=fi,g^c=gi,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, 357031);