#507505

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, 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

480 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=eg,c^2=fhi,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

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, 414701);