#490309

order	512
nilpotency class	4
isoclinism	isoclinic class of #490133
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	#62264	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#436 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

702 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 702 conjugacy classes of subgroups — too many to draw. Only its 136 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 — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,d^2=gh,e^2=gi,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=dghi,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

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397505);