#513480

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁷×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂⁴
commutator	C₂²	C₂³×C₄²
Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3654 conjugacy classes of subgroups; only the 1962 normal subgroups are shown.

normal subgroups only

This group has 3654 conjugacy classes of subgroups — too many to draw. Only its 1962 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) = 512

A minimal faithful permutation representation — 5 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=f,d^2=fh,e^2=fg,b^a=bf,c^a=cg,d^b=df

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420676);