#151088

order	512
nilpotency class	3
isoclinism	isoclinic class of #447
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₁₆	C₂×D₄
commutator	C₂×C₄	C₂²×C₁₆
Frattini, cosocle	C₄×C₁₆	C₂³
socle	C₂²	D₄×C₁₆
prime radical	C₂×D₄	C₂×C₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₄×C₃₂ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

142 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 142 conjugacy classes of subgroups — too many to draw. Only its 74 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

(2,5)(3,7)(6,8)(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);(1,2,4,6)(3,8,7,5)(10,11)(17,20)(18,21)(19,22)(30,33)(31,34)(32,35)(39,40);(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)

Polycyclic presentation:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58284);