#145216

order	512
nilpotency class	4
isoclinism	isoclinic class of #145216
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2034
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#1687	C₂
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#436 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

322 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 62 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) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|c^2=dh,d^2=g,e^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=cf,d^b=dg,e^b=ei,d^c=di,e^c=eghi,f^c=fg,h^c=hi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 52412);