#151877

order	512
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₆ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

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

254 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59073);