#134419

order	512
nilpotency class	4
isoclinism	isoclinic class of #124651
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#327
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

444 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 100 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)(7,16,18,33)(8,21,23,32)(9,14)(10,26)(11,15)(12,17)(13,24,30,25)(19,29,20,36)(22,31)(27,34)(28,35);(1,2,3,4);(5,7,11,18)(6,12,15,28)(8,22,23,34)(9,20,10,19)(13,29,30,36)(14,31,26,27)(16,32,33,21)(17,24,35,25)

Polycyclic presentation:

2^9|a^2=f,b^2=d,c^2=i,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eg,h^a=hi,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, 41615);