#147192

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×Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

505 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=hi,e^2=i,f^2=i,g^2=h,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, 54388);