#489401

order	512
nilpotency class	4
isoclinism	isoclinic class of #489398
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#62425	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2965 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

765 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 765 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=g,e^2=gi,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=dgi,e^a=eg,f^a=fhi,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,d^c=di,e^c=eh,f^c=fhi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396597);