#497755

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂³×C₄

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	C₂⁴	#131
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₂²⋊C₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

551 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

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, 404951);