#507487

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	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

532 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=g,c^2=fh,d^2=e,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,d^c=di

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, 414683);