#145896

order	512
nilpotency class	4
isoclinism	isoclinic class of #143214
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#447
commutator, Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1770 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

271 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 271 conjugacy classes of subgroups — too many to draw. Only its 88 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) = 512

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

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

Polycyclic presentation:

2^9|a^2=d,b^2=dh,c^2=dhi,e^2=gi,f^2=hi,g^2=i,b^a=bd,c^a=ce,e^a=eg,f^a=fh,g^a=gi,c^b=cf,e^b=eh,f^b=fhi,e^c=eg,f^c=fhi,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, 53092);