#406020

order	512
nilpotency class	3
isoclinism	isoclinic class of #401009
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

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

1074 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=fi,f^c=fi,g^c=gi,h^c=hi,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, 313216);