#150326

order	512
nilpotency class	5
isoclinism	isoclinic class of #7396
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#353
commutator	C₂×M₁₆	C₂²×C₄
Frattini, cosocle	#407	C₂³
socle	C₂	#11726
prime radical	#23794	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #2929 ◁ G
LECS	G ▷ #407 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

345 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 345 conjugacy classes of subgroups — too many to draw. Only its 54 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57522);