#396834

order	512
nilpotency class	3
isoclinism	isoclinic class of #396834
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14739
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

974 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)(3,11)(4,15)(5,17)(9,24)(10,20)(12,27)(13,14)(18,22)(19,25)(21,31)(26,28);(1,2,5,7,6,10,17,20)(3,8,11,22,14,23,13,18)(4,9,16,19,15,25,29,24)(12,21,26,32,28,31,27,30);(1,3)(2,8)(4,12)(5,13)(6,14)(7,18)(9,21)(10,23)(11,17)(15,28)(16,27)(19,30)(20,22)(24,32)(25,31)(26,29);(1,4,5,16,6,15,17,29)(2,9,7,19,10,25,20,24)(3,12,13,27,14,28,11,26)(8,21,18,30,23,31,22,32)

Polycyclic presentation:

2^9|b^2=f,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,g^b=gi,f^c=fi,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, 304030);