#403853

order	512
nilpotency class	3
isoclinism	isoclinic class of #394747
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2398
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1126 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)(5,6,10,13,11,16,9,15)(7,17,19,27,20,26,24,12)(8,14,23,28,21,30,22,29)(18,31,33,36,32,35,34,25);(5,7,9,19,11,20,10,24)(6,12,13,26,16,27,15,17)(8,18,22,33,21,32,23,34)(14,25,28,35,30,36,29,31);(6,13,16,15)(8,21)(12,17,27,26)(14,29,30,28)(18,32)(22,23)(25,35,36,31)(33,34);(2,4)(5,8,9,22,11,21,10,23)(6,14,15,29,16,30,13,28)(7,18,19,33,20,32,24,34)(12,25,17,31,27,36,26,35)

Polycyclic presentation:

2^9|a^2=fg,b^2=eg,c^2=g,d^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,d^c=di

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, 311049);