#142288

order	512
nilpotency class	4
isoclinism	isoclinic class of #142246
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	#33520	C₂
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#424 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ 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) = 128

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

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

Polycyclic presentation:

2^9|c^2=g,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=ei,g^b=gi,d^c=dh,e^c=ei,f^c=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 49484);