#151206

order	512
nilpotency class	5
isoclinism	isoclinic class of #151204
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

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

normal subgroups only

This group has 320 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fh,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,c^b=ch,d^b=dfi,e^b=egh,f^b=fi,g^b=gi,h^b=hi,d^c=dg,e^c=ei,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, 58402);