#493155

order	512
nilpotency class	4
isoclinism	isoclinic class of #3415
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator	C₈	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

778 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efhi,e^2=hi,f^2=hi,h^2=i,b^a=be,d^a=di,e^b=eh,f^b=fh,h^b=hi,d^c=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400351);