#507535

order	512
nilpotency class	4
isoclinism	isoclinic class of #499891
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

556 conjugacy classes of subgroups.

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 414731);