#104634

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, prime radical	C₂³	#392
commutator, Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₄³ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

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

normal subgroups only

This group has 435 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)(3,5)(4,7)(6,8)(9,10,14,18)(11,19,12,16)(13,22,23,17)(15,21,24,20)(30,39)(33,40);(3,8)(5,6)(9,11,14,12)(10,15,18,24)(13,21,23,20)(16,22,19,17)(25,26)(27,33)(28,36)(29,31)(30,35)(32,38)(34,40)(37,39);(2,5)(3,8)(6,7)(9,12,14,11)(10,16,18,19)(13,20,23,21)(15,17,24,22)(25,27)(26,30)(28,37)(29,34)(31,39)(32,40)(33,36)(35,38)

Polycyclic presentation:

2^9|a^2=g,b^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, 11830);