#498193

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	#525	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₂²⋊C₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

532 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=e,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,f^a=fi,d^b=dei,d^c=di,f^c=fi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 405389);