#492923

order	512
nilpotency class	2
isoclinism	isoclinic class of H(3,ℤ/4)
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄×C₈	C₄²
commutator	C₄	C₂²×C₄×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×#97
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

796 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)(9,10,14,20,15,22,30,37,16,21,29,36,13,19,12,18)(11,23,27,39,28,17,24,34,25,35,31,32,26,33,40,38);(2,5)(6,8)(9,11)(10,17,21,33)(12,24)(13,26)(14,31)(15,28)(16,25)(18,32,37,39)(19,23,22,35)(20,38,36,34)(27,29)(30,40);(1,3,4,7)(2,6,5,8)(9,12,13,29,16,30,15,14)(10,18,19,36,21,37,22,20)(11,24,26,27,25,40,28,31)(17,32,23,34,33,39,35,38);(11,25)(17,33)(23,35)(24,40)(26,28)(27,31)(32,39)(34,38)

Polycyclic presentation:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400119);