#486718

order	512
nilpotency class	4
isoclinism	isoclinic class of #486299
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	#62249	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

753 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 753 conjugacy classes of subgroups — too many to draw. Only its 132 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=ghi,e^2=gi,g^2=i,b^a=be,c^a=cf,d^a=dgi,e^a=eg,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,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, 393914);