#495971

order	512
nilpotency class	3
isoclinism	isoclinic class of #447
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₁₆	C₂×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	C₂×#41	C₂²×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

542 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)(3,8)(4,6)(5,7)(9,10,14,18,13,19,28,36,15,20,30,37,29,17,33,35)(11,21,23,26,24,32,38,27,25,34,39,40,22,16,31,12);(1,3,5,4)(2,6,7,8);(9,11)(10,16)(12,18)(13,24)(14,23)(15,25)(17,34)(19,21)(20,32)(22,29)(26,36)(27,37)(28,38)(30,39)(31,33)(35,40);(1,4,5,3)(2,6,7,8)(9,12,14,21,13,26,28,32,15,27,30,34,29,40,33,16)(10,11,18,23,19,24,36,38,20,25,37,39,17,22,35,31)

Polycyclic presentation:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403167);