#152791

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

Aliases

No aliases for this group.

Subgroups

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

200 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 44 points:

(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14,18,23,19,24,32,38,20,15,25,27,33,28,39,41,17,22,30,36,31,37,42,44,16,26,21,34,29,40,35,43);(6,9)(10,12)(13,15,18,27,19,28,32,41,20,22,25,36,33,37,39,44,17,26,30,34,31,40,42,43,16,14,21,23,29,24,35,38);(2,4)(5,7,8,11)(6,10,9,12)(13,16,17,20)(14,15,22,26)(18,21,30,25)(19,29,31,33)(23,27,36,34)(24,28,37,40)(32,35,42,39)(38,41,44,43)

Polycyclic presentation:

2^9|a^2=f,b^2=f,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, 59987);