#145530

order	512
nilpotency class	4
isoclinism	isoclinic class of #143951
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1667
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#2246	C₂
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

380 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 380 conjugacy classes of subgroups — too many to draw. Only its 78 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:

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

Polycyclic presentation:

2^9|c^2=d,d^2=gi,e^2=h,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fh,g^a=gi,c^b=cf,d^b=dg,e^b=eh,f^b=fh,g^b=gi,e^c=egh,f^c=fgh,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, 52726);