#479817

order	512
nilpotency class	3
isoclinism	isoclinic class of #479233
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2785
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17262
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

613 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387013);