#510186

order	512
nilpotency class	4
isoclinism	isoclinic class of #510185
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#31374
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3067	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

411 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=f,c^2=e,d^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=defh,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gh,h^b=hi,g^c=ghi,h^c=hi,f^d=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 417382);