#147223

order	512
nilpotency class	4
isoclinism	isoclinic class of #11651
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#102
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	C₄×#102
prime radical	C₂⁴	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

552 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)(5,6,10,14,11,15,27,16)(7,17,20,34,22,35,21,28)(8,13,23,31,24,32,9,25)(12,26,29,18,30,36,33,19);(5,7)(6,12,15,30)(8,18)(9,26)(10,20)(11,22)(13,28,32,34)(14,29,16,33)(17,25,35,31)(19,24)(21,27)(23,36);(2,4)(5,8,10,23,11,24,27,9)(6,13,14,31,15,32,16,25)(7,18,20,36,22,19,21,26)(12,28,29,17,30,34,33,35)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gi,h^a=hi,d^b=dhi,h^b=hi

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, 54419);