#120174

order	512
nilpotency class	3
isoclinism	isoclinic class of #11234
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#391
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1441
prime radical	C₂⁴	#117
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

552 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 27370);