#100224

order	512
nilpotency class	3
isoclinism	isoclinic class of #9519
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

432 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 98 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=g,d^2=g,e^2=g,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,e^b=eh,d^c=dh,e^c=eg,f^c=fh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7420);