#144218

order	512
nilpotency class	4
isoclinism	isoclinic class of #144216
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1999
commutator, Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁵	D₈
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×M₁₆ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

629 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 629 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=gi,g^2=i,b^a=bd,c^a=ce,d^a=dg,f^a=fhi,g^a=gi,c^b=cf,d^b=dg,e^b=eh,g^b=gi,d^c=di

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51414);