#151470

order	512
nilpotency class	5
isoclinism	isoclinic class of #151256
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2196
commutator, Frattini, cosocle	#390	C₂³
prime radical	#30551	C₂
LCS	G ▷ #390 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ #390 ◁ G
LECS	G ▷ #390 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 396 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) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58666);