#474857

order	512
nilpotency class	3
isoclinism	isoclinic class of #455110
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

930 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 930 conjugacy classes of subgroups — too many to draw. Only its 182 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) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=egh,c^2=eg,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=di,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 382053);