#145192

order	512
nilpotency class	4
isoclinism	isoclinic class of #145149
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

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

normal subgroups only

This group has 313 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=dh,d^2=i,e^2=g,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,f^a=fi,c^b=cf,d^b=di,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, 52388);