#147338

order	512
nilpotency class	4
isoclinism	isoclinic class of #2199
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	B(4,𝔽₂)
commutator	C₂×D₄	C₂×C₄²
Frattini, prime radical, cosocle	C₂³×D₄	C₂³
LCS	G ▷ C₂×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴ ◁ C₂³×D₄ ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂² ▷ C₂ ▷ C₁

992 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 992 conjugacy classes of subgroups — too many to draw. Only its 162 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:

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

Polycyclic presentation:

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

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, 54534);