#397900

order	512
nilpotency class	3
isoclinism	isoclinic class of #394289
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1895 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:

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

Polycyclic presentation:

2^9|b^2=e,c^2=fi,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,h^d=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, 305096);