#361454

order	512
nilpotency class	3
isoclinism	isoclinic class of #361285
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2290
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#13899
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1399 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 268650);