#6344309

order	512
nilpotency class	3
isoclinism	isoclinic class of #576
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₈ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	C₂²×D₄
commutator	C₄	C₂⁴×C₈
Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂³×C₄⋊C₄
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2100 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,e^2=h,f^2=h,g^2=i,b^a=bf,f^a=fh,d^c=dh

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6251505);