#104914

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

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

normal subgroups only

This group has 295 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=dh,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, 12110);