#501768

order	512
nilpotency class	4
isoclinism	isoclinic class of #501768
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24836
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3067	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

523 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408964);