#146826

order	512
nilpotency class	3
isoclinism	isoclinic class of #471
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#121
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1797
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

600 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54022);