#145202

order	512
nilpotency class	4
isoclinism	isoclinic class of #145147
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2034
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₈	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

304 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 76 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,c^2=dgh,d^2=i,e^2=g,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,c^b=cf,d^b=di,e^b=ei,f^b=fh,h^b=hi,d^c=di,e^c=egi,f^c=fh,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52398);