#145826

order	512
nilpotency class	4
isoclinism	isoclinic class of #145796
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2058
commutator, Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#336 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 387 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) = 24
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=d,b^2=d,c^2=dgh,e^2=g,f^2=i,g^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fhi,g^a=gi,c^b=cf,e^b=eh,d^c=di,e^c=eg,f^c=fi,g^c=gi

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, 53022);