#479795

order	512
nilpotency class	3
isoclinism	isoclinic class of #479795
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#17254
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

618 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fh,c^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=def,f^b=fi,g^b=gi,h^b=hi,d^c=de,e^c=ei,e^d=ei,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, 386991);