#489545

order	512
nilpotency class	4
isoclinism	isoclinic class of #489398
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂×#41	D₈
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2965 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

657 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 657 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,d^2=g,e^2=gi,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=dgi,e^a=eg,f^a=fhi,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,d^c=di,e^c=eh,f^c=fhi

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, 396741);