#104932

order	512
nilpotency class	3
isoclinism	isoclinic class of #104917
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₄

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

306 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 76 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)(11,15)(12,14)(13,17)(16,20)(18,21)(19,22)(23,24)(25,26,30,34)(27,28,32,38)(29,35,40,33)(31,39,37,36);(1,2,3,7)(4,8,5,6)(10,14)(11,18)(12,20)(15,22)(16,17)(19,24)(25,27)(26,31)(28,35)(29,39)(30,32)(33,38)(34,37)(36,40);(2,6)(4,5)(7,8)(9,11,12,18,13,19,20,24)(10,15,16,23,17,22,14,21)(25,28,29,37,30,38,40,31)(26,32,33,39,34,27,35,36)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=dh,d^2=g,e^2=h,f^2=ghi,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=fghi,e^c=egh,f^c=fhi

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, 12128);