#326517

order	512
nilpotency class	3
isoclinism	isoclinic class of #326513
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2563
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

485 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 485 conjugacy classes of subgroups — too many to draw. Only its 142 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 — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=egh,d^2=ei,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dei,e^a=eh,f^a=fh,c^b=cg,f^c=fi,g^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 233713);