#147193

order	512
nilpotency class	4
isoclinism	isoclinic class of #13068
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1893
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10880
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

513 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,6,13,8,14,7,12)(3,15,18,44,21,35,20,43)(4,22,25,38,26,41,24,10)(5,27,28,40,30,39,29,11)(9,31,34,19,36,51,33,17)(16,45,48,59,50,63,47,42)(23,54,56,62,57,37,55,53)(32,46,52,58,60,49,61,64);(1,3,5,19,6,18,28,51,8,21,30,17,7,20,29,31)(2,9,11,35,12,33,39,44,14,36,40,15,13,34,27,43)(4,16,23,49,25,48,56,64,26,50,57,46,24,47,55,58)(10,32,37,59,41,61,62,45,38,60,54,42,22,52,53,63);(1,4,6,25,8,26,7,24)(2,10,13,22,14,38,12,41)(3,16,18,48,21,50,20,47)(5,23,28,56,30,57,29,55)(9,32,34,52,36,60,33,61)(11,37,27,53,40,54,39,62)(15,42,44,45,35,59,43,63)(17,46,31,58,19,49,51,64)

Polycyclic presentation:

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

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, 54389);