#495915

order	512
nilpotency class	3
isoclinism	isoclinic class of #3008
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 ◁ C₂²×M₃₂ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

557 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403111);