#495795

order	512
nilpotency class	3
isoclinism	isoclinic class of #3008
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

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	#3035	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

650 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^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, 402991);