#134320

order	512
nilpotency class	4
isoclinism	isoclinic class of #11499
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#99
commutator, prime radical	C₂⁴	C₂²×C₈
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂²	#1459
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

404 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 110 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dh,c^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gi,e^c=ei,f^c=fg,f^e=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41516);