#147100

order	512
nilpotency class	4
isoclinism	isoclinic class of #2221
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#466
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

550 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ehi,d^2=hi,e^2=hi,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gh,h^a=hi,d^b=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54296);