#104807

order	512
nilpotency class	3
isoclinism	isoclinic class of #104789
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#393
commutator, Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#120
LCS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

458 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 84 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)(4,7)(6,8)(9,10)(11,18,20,24)(12,16)(13,22)(14,17)(15,21,23,19)(25,26)(27,30)(28,35,37,31)(29,32)(33,38,36,40)(34,39);(11,19,20,21)(15,18,23,24)(25,27,29,34)(26,30,32,39)(28,36)(31,40)(33,37)(35,38);(2,5)(3,8)(6,7)(9,11)(10,15)(12,19)(13,21)(14,20)(16,24)(17,23)(18,22)(25,28,29,37)(26,31,32,35)(27,33,34,36)(30,38,39,40)

Polycyclic presentation:

2^9|a^2=d,b^2=di,c^2=i,e^2=gh,f^2=gi,b^a=bd,c^a=ce,e^a=eh,f^a=fgi,c^b=cf,e^b=ei,f^b=fi,d^c=dg,e^c=egh,f^c=fgi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 12003);