#145618

order	512
nilpotency class	4
isoclinism	isoclinic class of #145618
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2044
commutator, Frattini, cosocle	C₂×C₄×D₄	C₂³
prime radical	#31056	C₂
LCS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

542 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 52 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) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 24 points:

(2,5)(6,8)(9,10)(11,18)(12,14)(13,17)(15,19)(16,21)(20,23)(22,24);(3,7)(10,14)(11,19)(12,21)(15,23)(16,17)(18,22);(1,2)(3,5)(4,6)(7,8)(9,11,12,20,13,19,21,24)(10,15,16,23,17,18,14,22)

Polycyclic presentation:

2^9|c^2=d,d^2=g,b^a=bd,c^a=ce,d^a=dg,f^a=fhi,c^b=cf,d^b=dg,e^b=eh,e^c=eg,f^c=fg,h^c=hi,f^e=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52814);