#455391

order	512
nilpotency class	3
isoclinism	isoclinic class of #455391
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16263
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

1126 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:

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

Polycyclic presentation:

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

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, 362587);