#109078

order	512
nilpotency class	3
isoclinism	isoclinic class of #8794
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#382
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₄⋊#95
prime radical	C₂⁴	C₄⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

470 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 190 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) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,4,9,6,10,5,8)(3,11,13,7,12,16,14,15)(17,18,22,27,23,26,21,37)(19,28,30,40,31,24,38,41)(20,32,35,43,36,46,34,25)(29,44,45,42,33,47,48,39);(1,3,5,14,6,12,4,13)(2,7,9,16,10,15,8,11)(17,19)(18,24)(20,33)(21,30)(22,38)(23,31)(25,42)(26,28)(27,40)(29,36)(32,44)(34,48)(35,45)(37,41)(39,43)(46,47);(3,12)(7,15)(11,16)(13,14)(17,20,21,34,23,36,22,35)(18,25,27,32,26,43,37,46)(19,29,30,45,31,33,38,48)(24,39,40,47,28,42,41,44)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 16274);