#147170

order	512
nilpotency class	3
isoclinism	isoclinic class of #8603
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#117
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1770
prime radical	C₂⁴	#131
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×M₃₂ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

514 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=ei,f^b=fi,d^c=di,e^c=ei,g^c=gi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54366);