#145049

order	512
nilpotency class	4
isoclinism	isoclinic class of #145004
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

312 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 74 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=dg,e^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fhi,g^a=gi,c^b=cf,d^b=di,e^b=eh,f^b=fhi,d^c=di,e^c=egi,f^c=fhi,g^c=gi

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, 52245);