#146848

order	512
nilpotency class	3
isoclinism	isoclinic class of #11247
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#387
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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,e^2=i,f^2=h,g^2=h,h^2=i,b^a=bd,c^a=ce,g^a=gi,c^b=ch,d^b=di,e^c=ei,f^c=fi,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, 54044);