#104630

order	512
nilpotency class	3
isoclinism	isoclinic class of #103005
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#392
commutator, Frattini, cosocle	C₄³	C₂³
prime radical	C₂×#127	D₄
LCS	G ▷ C₄³ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

466 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 72 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)(9,10,13,17)(11,15,19,24)(12,14,20,16)(18,23,21,22)(30,39)(33,40);(3,8)(5,6)(10,14)(12,20)(15,23)(16,17)(18,21)(22,24)(25,26)(27,33)(28,36)(29,31)(30,35)(32,38)(34,40)(37,39);(2,5)(3,8)(6,7)(9,11,13,19)(10,15,17,24)(12,21,20,18)(14,22,16,23)(25,27)(26,30)(28,37)(29,34)(31,39)(32,40)(33,36)(35,38)

Polycyclic presentation:

2^9|a^2=g,c^2=g,d^2=g,e^2=i,f^2=gh,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=fgh,d^c=dh,e^c=ei,f^c=fgh

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, 11826);