#489110

order	512
nilpotency class	4
isoclinism	isoclinic class of #488916
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	C₄×C₈	C₂⁴
prime radical	#532	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

666 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 666 conjugacy classes of subgroups — too many to draw. Only its 146 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 396306);