#488849

order	512
nilpotency class	4
isoclinism	isoclinic class of #488699
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	C₂²×C₈	C₂⁴
prime radical	#466	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#446 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 678 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:

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

Polycyclic presentation:

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

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, 396045);