#486978

order	512
nilpotency class	4
isoclinism	isoclinic class of #486811
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

700 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 116 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=i,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=hi,c^b=ci,d^b=di,e^b=eg,f^b=fhi,d^c=dgi,e^c=eh,f^e=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 394174);