#137219

order	512
nilpotency class	4
isoclinism	isoclinic class of #12304
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1616
commutator	C₂×M₁₆	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 350 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=hi,d^2=g,f^2=gi,g^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,d^b=dg,e^b=ehi,f^b=fgi,g^b=gi,d^c=dh,e^c=ei,f^c=fh,e^d=ei,f^e=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44415);