#138573

order	512
nilpotency class	4
isoclinism	isoclinic class of #12465
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1677
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#135	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

313 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 313 conjugacy classes of subgroups — too many to draw. Only its 96 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,d^2=ghi,e^2=hi,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dh,e^b=ehi,f^b=fhi,h^b=hi,d^c=dhi,e^c=eh,f^c=fh,h^c=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45769);