#142286

order	512
nilpotency class	4
isoclinism	isoclinic class of #142248
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	C₂³×D₈	C₂²
LCS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#424 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

638 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 638 conjugacy classes of subgroups — too many to draw. Only its 72 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)(9,10,14,19)(11,20,23,16)(12,15,27,31)(13,28,26,18)(17,34,30,24)(21,33,38,36)(22,29,25,40)(32,37,35,39);(5,8)(10,15)(11,21)(12,24)(13,26)(16,32)(17,18)(19,31)(20,35)(22,25)(23,38)(27,34)(28,30)(29,33)(36,40);(1,3)(2,5)(4,7)(6,8)(9,11,13,21,14,23,26,38)(10,16,18,33,19,20,28,36)(12,25,24,39,27,22,34,37)(15,29,30,32,31,40,17,35)

Polycyclic presentation:

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

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, 49482);