#144567

order	512
nilpotency class	4
isoclinism	isoclinic class of #144565
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2023
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	#3169	C₂²
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#97 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 298 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)(3,10,12,16)(4,8)(5,14)(6,9)(7,13,15,11)(17,18,19,22)(20,23,26,30)(21,29,27,25)(24,32,31,28);(3,11,12,13)(7,10,15,16)(18,22)(20,26)(25,29)(28,32);(1,3)(2,7)(4,11)(5,13)(6,12)(8,16)(9,15)(10,14)(18,23)(20,28)(21,27)(22,30)(24,25)(26,32)(29,31)

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 51763);