#507154

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

499 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)(11,14)(12,13)(15,16)(17,18,21,26)(19,29,32,24)(20,25,23,28)(22,27,30,38)(31,41,34,39)(33,45,46,40)(35,42,37,44)(36,48,47,43);(1,3,4,8)(2,5,6,7)(17,19,30,46,20,34,37,36,21,32,22,33,23,31,35,47)(18,24,38,45,25,41,44,43,26,29,27,40,28,39,42,48);(10,13)(14,16)(17,19,21,32)(18,24,26,29)(20,31,23,34)(22,36,30,47)(25,39,28,41)(27,43,38,48)(33,35,46,37)(40,42,45,44);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,20,21,23)(18,25,26,28)(19,31,32,34)(22,35,30,37)(24,39,29,41)(27,42,38,44)(33,36,46,47)(40,43,45,48)

Polycyclic presentation:

2^9|a^2=e,b^2=eg,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, 414350);