#151333

order	512
nilpotency class	5
isoclinism	isoclinic class of #151330
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#12186
commutator, Frattini, cosocle	#407	C₂³
prime radical	#30434	C₂
LCS	G ▷ #407 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #407 ◁ G
LECS	G ▷ #407 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

417 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 417 conjugacy classes of subgroups — too many to draw. Only its 36 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=fgi,d^2=fi,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=df,e^a=ei,f^a=fi,g^a=gh,d^b=df,e^b=egh,f^b=fi,g^b=gi,h^b=hi,d^c=dg,e^c=eh,f^c=fi,g^c=gi,e^d=ehi,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, 58529);