#507693

order	512
nilpotency class	4
isoclinism	isoclinic class of #501046
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 ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

513 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)(4,5)(6,8)(11,15)(13,16)(17,18,20,26,32,40,44,38,22,25,33,39,21,27,31,24)(19,23,28,36,42,47,48,45,30,35,43,46,29,37,41,34);(2,5)(6,7)(9,10,12,14)(11,16,15,13)(17,19,22,30)(18,23,25,35)(20,28,33,43)(21,29,32,42)(24,34,38,45)(26,36,39,46)(27,37,40,47)(31,41,44,48);(9,11,12,15)(10,13,14,16)(18,24)(20,31)(21,32)(23,34)(25,38)(26,27)(28,41)(29,42)(33,44)(35,45)(36,37)(39,40)(43,48)(46,47);(1,3,4,8)(2,6,5,7)(18,25)(23,35)(24,38)(26,39)(27,40)(34,45)(36,46)(37,47)

Polycyclic presentation:

2^9|a^2=fg,b^2=gi,c^2=g,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,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, 414889);