#137213

order	512
nilpotency class	4
isoclinism	isoclinic class of #12298
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1613
commutator	C₂×M₁₆	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂⁵	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

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

normal subgroups only

This group has 482 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)(5,6,10,16,25,13,27,32,11,15,24,34,8,17,9,14)(7,18,22,12,19,31,28,36,21,35,26,30,23,33,20,29);(1,2,3,4)(5,7)(6,12,15,30)(8,19)(9,26)(10,28)(11,21)(13,29,17,36)(14,33,32,31)(16,35,34,18)(20,24)(22,27)(23,25);(5,8,11,25)(6,13,15,17)(7,19,21,23)(9,24,27,10)(12,29,30,36)(14,34,32,16)(18,31,35,33)(20,22,28,26)

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=i,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,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, 44409);