#145184

order	512
nilpotency class	4
isoclinism	isoclinic class of #145129
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2033
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

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

normal subgroups only

This group has 310 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)(11,20)(12,15)(13,18)(14,19)(16,23)(17,25)(21,37)(22,32)(24,29)(26,30)(27,36)(28,35)(31,39)(33,38)(34,40);(2,5)(6,7)(10,15)(11,21)(12,25)(13,26)(16,31)(17,19)(18,35)(20,36)(22,39)(23,38)(24,27)(28,40)(29,33)(32,37);(2,6)(4,8)(5,7)(9,11,12,22,14,23,25,24)(10,16,17,32,19,20,15,29)(13,27,26,38,28,39,40,21)(18,31,34,33,35,36,30,37)

Polycyclic presentation:

2^9|a^2=g,c^2=dg,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,f^b=fh,h^b=hi,e^c=egi,f^c=fhi,h^c=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52380);