#134462

order	512
nilpotency class	4
isoclinism	isoclinic class of #124681
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1514
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

401 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 401 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dg,c^2=i,d^2=i,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,h^a=hi,e^b=ei,d^c=di,e^c=eh,f^c=fghi,g^c=gi,h^c=hi,f^e=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41658);