#104616

order	512
nilpotency class	3
isoclinism	isoclinic class of #104565
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#392
commutator, Frattini, cosocle	C₄³	C₂³
prime radical	C₂×#127	D₄
LCS	G ▷ C₄³ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

454 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=gh,c^2=gi,d^2=g,e^2=i,f^2=h,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=fh,d^c=dh,e^c=ei,f^c=fh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 11812);