#491310

order	512
nilpotency class	4
isoclinism	isoclinic class of #491245
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	#33773	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

423 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 423 conjugacy classes of subgroups — too many to draw. Only its 126 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:

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

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,e^a=eg,f^a=fh,g^a=gi,d^b=dh,d^c=di,f^c=fh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398506);