#460670

order	512
nilpotency class	3
isoclinism	isoclinic class of #460670
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16369
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

913 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 367866);