#507678

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

564 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)(11,15)(12,13)(14,16)(17,18,24,38,20,27,37,41,23,30,21,28,22,29,35,48)(19,25,39,45,34,47,36,26,33,44,32,43,31,42,40,46);(1,3,4,8)(2,5,6,7)(10,13)(11,16)(17,19)(18,25)(20,34)(21,32)(22,31)(23,33)(24,39)(26,41)(27,47)(28,43)(29,42)(30,44)(35,40)(36,37)(38,45)(46,48);(9,11,12,16)(10,14,13,15)(17,19)(18,26)(20,31)(21,36)(22,34)(23,33)(24,40)(25,41)(27,45)(28,42)(29,43)(30,46)(32,37)(35,39)(38,47)(44,48);(2,6)(5,7)(10,13)(14,15)(17,20,23,22)(18,27,30,29)(19,31,33,34)(21,35,24,37)(25,42,44,47)(26,45,46,43)(28,48,38,41)(32,36,39,40)

Polycyclic presentation:

2^9|a^2=fgi,b^2=eg,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, 414874);