#142326

order	512
nilpotency class	4
isoclinism	isoclinic class of #142228
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2021
commutator, Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

569 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 569 conjugacy classes of subgroups — too many to draw. Only its 70 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=g,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fhi,c^b=cf,d^b=dg,f^b=fi,h^b=hi,d^c=dh,e^c=ei,f^c=fi,f^d=fi

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, 49522);