#486108

order	512
nilpotency class	4
isoclinism	isoclinic class of #486091
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	#534	C₂³
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

780 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 780 conjugacy classes of subgroups — too many to draw. Only its 118 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)(3,9)(4,12)(5,7)(6,11)(8,14)(10,16)(13,15)(17,18)(19,29)(20,25)(21,26)(22,27)(23,28)(24,31)(30,41)(32,44)(33,42)(34,43)(35,39)(36,48)(37,45)(38,46)(40,47);(1,3)(2,6)(4,13)(5,10)(7,15)(8,16)(9,12)(11,14)(17,19,23,21)(18,24,28,39)(20,32,34,30)(22,31,37,35)(25,40,43,38)(26,27,29,45)(33,47,36,46)(41,42,44,48);(17,20,23,34)(18,25,28,43)(19,30,21,32)(22,36,37,33)(24,38,39,40)(26,44,29,41)(27,42,45,48)(31,47,35,46);(2,7)(3,10)(8,12)(11,13)(17,21,23,19)(18,26,28,29)(20,30,34,32)(22,35,37,31)(24,27,39,45)(25,41,43,44)(33,46,36,47)(38,48,40,42)

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 393304);