#507725

order	512
nilpotency class	4
isoclinism	isoclinic class of #499897
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

543 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=i,d^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cg,d^b=di,f^c=fh,h^c=hi

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, 414921);