#491142

order	512
nilpotency class	4
isoclinism	isoclinic class of #491091
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#33733	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

419 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 419 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,9)(4,12)(5,7)(6,10)(8,14)(11,16)(13,15)(18,24)(19,30)(20,22)(21,25)(26,42)(27,39)(28,38)(29,40)(31,46)(32,48)(33,34)(35,36)(37,44)(45,47);(17,18,25,39,22,28,36,40,23,29,35,38,20,27,21,24)(19,31,41,46,30,45,48,37,33,26,43,42,34,44,32,47);(1,3,5,11)(2,6,7,15)(4,13,14,10)(8,16,12,9)(17,19,23,33)(18,26,29,31)(20,34,22,30)(21,32,36,48)(24,37,40,47)(25,41,35,43)(27,45,28,44)(38,46,39,42);(2,7)(6,15)(8,12)(9,16)(17,20,23,22)(18,27,29,28)(19,30,33,34)(21,35,36,25)(24,38,40,39)(26,44,31,45)(32,41,48,43)(37,42,47,46)

Polycyclic presentation:

2^9|b^2=eg,c^2=h,d^2=i,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fh,g^a=gi,c^b=ci,d^c=di,f^c=fh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398338);