#497109

order	512
nilpotency class	4
isoclinism	isoclinic class of #483987
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂³	C₂×AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#97 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

447 conjugacy classes of subgroups.

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=efh,d^2=ei,e^2=i,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,h^a=hi,c^b=ce,d^b=di,f^b=fi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404305);