#104362

order	512
nilpotency class	3
isoclinism	isoclinic class of #104309
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#392
commutator, Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

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

normal subgroups only

This group has 495 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=g,d^2=g,e^2=i,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fgi,c^b=cf,d^b=dg,e^b=eghi,f^b=fg,d^c=dh,e^c=ei,f^c=fg

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 11558);