#399150

order	512
nilpotency class	3
isoclinism	isoclinic class of #399150
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14776
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

1246 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 306346);