#413892

order	512
nilpotency class	3
isoclinism	isoclinic class of #401009
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1407 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2)(3,4)(5,6)(7,17)(8,22)(9,13)(10,15)(11,16)(12,18)(14,24)(19,33)(20,27)(21,30)(23,29)(25,28)(26,35)(31,32)(34,36);(2,4)(5,7,11,21)(6,12,16,27)(8,19,25,35)(9,20,10,18)(13,17,15,30)(14,26,29,33)(22,36,28,31)(23,32,24,34);(6,13)(7,18)(9,10)(14,28)(15,16)(17,30)(19,32)(20,21)(22,29)(23,24)(31,36)(34,35);(5,8)(6,14,16,29)(7,19)(9,24)(10,23)(11,25)(12,26,27,33)(13,28,15,22)(17,31,30,36)(18,32)(20,34)(21,35)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 321088);