#145054

order	512
nilpotency class	4
isoclinism	isoclinic class of #143214
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#447
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₈	D₄
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#446 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

351 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 351 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|b^2=g,c^2=dg,d^2=i,e^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,f^a=fhi,g^a=gi,c^b=cf,e^b=eh,f^b=fhi,d^c=di,e^c=eg,f^c=fh,g^c=gi

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, 52250);