#507637

order	512
nilpotency class	4
isoclinism	isoclinic class of #501064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

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 ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

510 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)(17,18,24,39,20,34,38,48,21,28,22,29,23,27,36,42)(19,30,40,44,35,45,37,26,32,25,33,46,31,43,41,47);(1,3,4,8)(2,5,6,7)(10,13)(11,15)(17,19)(18,25)(20,35)(21,32)(22,33)(23,31)(24,40)(26,42)(27,45)(28,30)(29,44)(34,43)(36,41)(37,38)(39,46)(47,48);(9,11,12,15)(10,14,13,16)(17,19)(18,26)(20,31)(21,32)(22,37)(23,35)(24,41)(25,42)(27,46)(28,47)(29,43)(30,48)(33,38)(34,44)(36,40)(39,45);(2,6)(5,7)(10,13)(14,16)(17,20,21,23)(18,27,28,34)(19,31,32,35)(22,36,24,38)(25,43,30,45)(26,46,47,44)(29,48,39,42)(33,37,40,41)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 414833);