#146847

order	512
nilpotency class	3
isoclinism	isoclinic class of #11115
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#388
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1924
prime radical	C₂⁴	#131
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

474 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)(5,6,8,14,10,15,23,29,11,13,9,24,22,28,25,34)(7,16,19,31,21,30,33,36,18,12,20,27,17,26,32,35);(5,7,9,20,10,21,25,32,11,18,8,19,22,17,23,33)(6,12,14,27,15,26,29,35,13,16,24,31,28,30,34,36);(2,4)(6,13)(7,17,18,21)(8,9)(12,26,16,30)(15,28)(19,33,20,32)(23,25)(27,36,31,35)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54043);