#507296

order	512
nilpotency class	4
isoclinism	isoclinic class of #500839
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	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊C₄ ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

471 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=efgi,c^2=i,d^2=f,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,d^b=di,g^c=gh,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, 414492);