#146865

order	512
nilpotency class	4
isoclinism	isoclinic class of #11508
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#99
commutator	C₂²×C₄	C₂×C₄²
Frattini, prime radical, cosocle	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂² ▷ C₂ ▷ C₁

972 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 972 conjugacy classes of subgroups — too many to draw. Only its 158 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,e^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,c^b=ci,e^c=ei,f^c=fhi,f^e=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54061);