#146851

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	C₈	#388
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10798
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

463 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,d^2=i,e^2=i,f^2=h,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,c^b=ch,d^b=di,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, 54047);