#151316

order	512
nilpotency class	5
isoclinism	isoclinic class of #151316
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#12181
commutator, Frattini, cosocle	#423	C₂³
prime radical	#3253	C₂²
LCS	G ▷ #423 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #423 ◁ G
LECS	G ▷ #423 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

209 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 209 conjugacy classes of subgroups — too many to draw. Only its 38 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) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=fh,c^2=g,d^2=fi,e^2=hi,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=df,e^a=ehi,f^a=fi,g^a=gh,d^b=dfi,e^b=eghi,f^b=fi,h^b=hi,d^c=dg,e^c=ei,e^d=eh,g^d=gi

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58512);