#142229

order	512
nilpotency class	4
isoclinism	isoclinic class of #142228
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2021
commutator, Frattini, cosocle	C₂×#117	C₂³
prime radical	#18300	C₂
LCS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 760 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:

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

Polycyclic presentation:

2^9|c^2=i,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, 49425);