#145561

order	512
nilpotency class	4
isoclinism	isoclinic class of #144059
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1677
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#493	C₂²
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

288 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 288 conjugacy classes of subgroups — too many to draw. Only its 74 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)(5,6,10,14)(7,17)(8,11,23,25)(9,15,22,16)(12,27)(13,32,26,21)(18,29)(19,30)(20,31)(24,35)(28,33)(34,36);(6,11)(8,21)(9,22)(12,28)(13,16)(14,25)(15,26)(17,29)(18,33)(19,27)(23,32)(24,30)(31,36)(34,35);(2,4)(5,7,8,18,22,27,32,35,10,20,23,34,9,19,21,33)(6,12,13,29,15,17,25,28,14,30,26,36,16,31,11,24)

Polycyclic presentation:

2^9|a^2=h,c^2=d,d^2=gi,e^2=hi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fhi,g^a=gi,c^b=cf,d^b=dg,e^b=ehi,f^b=fh,g^b=gi,e^c=eghi,f^c=fgh,h^c=hi

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, 52757);