#505654

order	512
nilpotency class	4
isoclinism	isoclinic class of #500591
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#3253
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	#27626
prime radical	#3050	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

478 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 412850);