#99078

order	512
nilpotency class	3
isoclinism	isoclinic class of #98920
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

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

338 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 106 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:

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

Polycyclic presentation:

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

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, 6274);