#504677

order	512
nilpotency class	4
isoclinism	isoclinic class of #504676
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#26801
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#13310	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1765 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

509 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)(3,13)(4,5)(6,10)(7,23)(8,12)(9,15)(11,25)(14,24)(16,30)(17,18)(19,22)(20,29)(21,28)(26,27)(31,32);(1,3,5,15,8,17,6,21)(2,9,10,13,12,28,4,18)(7,24,20,32,25,26,22,16)(11,30,29,27,23,31,19,14);(1,4,5,2,8,10,6,12)(3,14,15,30,17,27,21,31)(7,11,20,29,25,23,22,19)(9,26,13,16,28,24,18,32);(2,4,12,10)(7,25)(9,18,28,13)(11,29,23,19)(14,30,27,31)(16,32)(20,22)(24,26)

Polycyclic presentation:

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

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, 411873);