#104905

order	512
nilpotency class	3
isoclinism	isoclinic class of #104897
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#394
commutator, Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₄³ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

305 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 305 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=d,d^2=g,e^2=h,f^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fi,c^b=cf,d^b=dg,e^b=ei,f^b=fhi,e^c=egh,f^c=fghi

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, 12101);