#99089

order	512
nilpotency class	3
isoclinism	isoclinic class of #98936
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#380
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

332 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 110 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) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dgh,c^2=h,d^2=g,e^2=i,f^2=ghi,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=cg,e^c=ei,f^c=fhi

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, 6285);