#104339

order	512
nilpotency class	3
isoclinism	isoclinic class of #104309
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#392
commutator, Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

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

normal subgroups only

This group has 537 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=i,d^2=g,e^2=i,f^2=g,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=fg,d^c=dh,e^c=ei,f^c=fg

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, 11535);