#474674

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

728 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 728 conjugacy classes of subgroups — too many to draw. Only its 198 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:

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

Polycyclic presentation:

2^9|a^2=f,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, 381870);