#146048

order	512
nilpotency class	4
isoclinism	isoclinic class of #146048
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2073
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₈	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 238 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:

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

Polycyclic presentation:

2^9|a^2=eh,c^2=d,e^2=gi,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,c^b=cf,f^b=fh,h^b=hi,e^c=eg,f^c=fh,h^c=hi

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, 53244);