#510219

order	512
nilpotency class	4
isoclinism	isoclinic class of #510219
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#31382
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#444 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

513 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=ef,c^2=e,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gh,h^b=hi,f^c=fi,g^c=gh,h^c=hi,f^d=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 417415);