#151329

order	512
nilpotency class	5
isoclinism	isoclinic class of #151326
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

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

normal subgroups only

This group has 395 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)(3,13,17,31)(4,9,18,10)(5,19,22,28)(6,23,21,11)(8,14,26,15)(16,25,24,29)(20,30,32,27);(1,3)(2,8)(4,14)(5,20)(6,24)(7,17)(9,25)(10,29)(11,30)(12,26)(13,19)(15,18)(16,21)(22,32)(23,27)(28,31);(1,4,6,22,7,18,21,5)(2,9,23,28,12,10,11,19)(3,14,24,32,17,15,16,20)(8,25,27,31,26,29,30,13)

Polycyclic presentation:

2^9|a^2=i,c^2=fgi,d^2=fi,f^2=i,b^a=bd,c^a=ce,d^a=df,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=ehi,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, 58525);