#491242

order	512
nilpotency class	4
isoclinism	isoclinic class of #491239
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	#19332	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

445 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 445 conjugacy classes of subgroups — too many to draw. Only its 128 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)(3,6)(4,7)(5,9)(8,12)(10,14)(11,13)(15,16)(17,18,23,27)(19,28,25,29)(20,32,33,43)(21,37,39,31)(22,40,38,26)(24,36,41,35)(30,34,42,48)(44,47,45,46);(1,3)(2,6)(4,11)(5,10)(7,13)(8,15)(9,14)(12,16)(17,19,29,18,22,31,21,26,23,25,28,27,38,37,39,40)(20,30,45,24,36,47,34,43,33,42,44,41,35,46,48,32);(2,7)(4,12)(6,13)(8,9)(11,16)(14,15)(17,20)(18,24)(19,30)(21,34)(22,36)(23,33)(25,42)(26,43)(27,41)(28,44)(29,45)(31,47)(32,40)(35,38)(37,46)(39,48);(3,10)(6,14)(11,16)(13,15)(20,33)(24,41)(30,42)(32,43)(34,48)(35,36)(44,45)(46,47)

Polycyclic presentation:

2^9|a^2=i,b^2=egi,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=dhi,d^c=di,f^c=fh

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398438);