#145048

order	512
nilpotency class	4
isoclinism	isoclinic class of #145002
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2033
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#452
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

308 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 76 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 — 3 generators on 40 points:

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

Polycyclic presentation:

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

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, 52244);