#106061

order	512
nilpotency class	3
isoclinism	isoclinic class of #8626
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#380
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 966 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) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 44 points:

(1,2)(3,4)(5,6,7,10,8,9,11,12)(13,14)(15,20,25,35)(16,27,28,38)(17,23)(18,32)(19,22)(21,29,37,31)(24,39)(26,34,30,36)(33,41)(40,44)(42,43);(2,4)(6,9)(7,11)(13,15,18,26,19,25,17,30)(14,20,23,36,22,35,32,34)(16,24,29,41,28,42,31,40)(21,33,38,39,37,44,27,43);(6,9)(7,11)(13,16)(14,21)(15,24)(17,31)(18,29)(19,28)(20,33)(22,37)(23,38)(25,42)(26,41)(27,32)(30,40)(34,43)(35,44)(36,39)

Polycyclic presentation:

2^9|a^2=f,b^2=g,e^2=i,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,g^a=gi,d^b=di,e^b=ei,f^b=fh,d^c=di,e^c=ei,f^c=fh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 13257);