#150161

order	512
nilpotency class	5
isoclinism	isoclinic class of #150157
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#357
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₄×C₁₆	C₂³
prime radical	C₂³	#357
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

213 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 213 conjugacy classes of subgroups — too many to draw. Only its 78 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,d^2=fhi,e^2=g,f^2=hi,h^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,c^b=cgi,d^b=dfi,e^b=ei,f^b=fh,h^b=hi

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, 57357);