#411835

order	512
nilpotency class	3
isoclinism	isoclinic class of #395169
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

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

2032 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)(11,23)(12,27)(13,17)(14,21)(15,20)(16,22)(18,29)(19,26)(24,38)(25,33)(28,37)(30,36)(31,35)(32,39)(34,40);(1,2,3,4)(5,6)(7,8)(9,11,14,13,16,26,15,25)(10,17,20,23,22,33,21,19)(12,24,29,28,30,39,31,40)(18,32,27,37,35,38,36,34);(6,8)(9,12)(10,18)(11,24)(13,28)(14,29)(15,31)(16,30)(17,32)(19,34)(20,27)(21,36)(22,35)(23,37)(25,40)(26,39)(33,38);(9,13,14,26,16,25,15,11)(10,19,20,17,22,23,21,33)(12,28,29,39,30,40,31,24)(18,34,27,32,35,37,36,38)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 319031);