#152807

order	512
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₁₆	C₂³
commutator, socle	C₂³	C₂²×C₁₆
prime radical	C₂⁴	C₂×C₁₆
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

242 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=d,f^2=g,g^2=h,h^2=i,b^a=bd,c^a=ce,c^b=ci

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, 60003);