#151202

order	512
nilpotency class	5
isoclinism	isoclinic class of #151200
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#12124
commutator, Frattini, cosocle	#407	C₂³
prime radical	C₂³×D₄	D₄
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

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

normal subgroups only

This group has 342 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

29 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58398);